Algorithmic algebraic combinatorics and Groebner bases

Algorithmic Algebraic Combinatorics and Gröbner Bases Mikhail Klin · Gareth A. Jones · Aleksandar Juriši´c · Mikhail ...

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Algorithmic Algebraic Combinatorics and Gröbner Bases

Mikhail Klin · Gareth A. Jones · Aleksandar Juriši´c · Mikhail Muzychuk · Ilia Ponomarenko Editors

Algorithmic Algebraic Combinatorics and Gröbner Bases

Mikhail Klin Ben-Gurion University of the Negev Department of Mathematics POB 653 84633 Beer Sheva Israel [email protected]

Mikhail Muzychuk Department of Mathematics and Computer Science Netanya Academic College University st. 1 42 365 Netanya Israel [email protected]

Gareth A. Jones School of Mathematics University of Southampton Southampton SO17 1BJ United Kingdom [email protected]

Ilia Ponomarenko Laboratory of Representation Theory and Computational Mathematics V.A. Steklov Mathematical Institute Fontanka 27 191023 St. Petersburg Russia [email protected]

Aleksandar Juriši´c Laboratory for Cryptography and Computer Security Faculty of Computer Science Tržaška cesta 25 1000 Ljubljana Slovenia [email protected]

ISBN 978-3-642-01959-3 e-ISBN 978-3-642-01960-9 DOI 10.1007/978-3-642-01960-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009927505 Mathematics Subject Classification (2000): 05E30, 13P10, 20B25, 68-04 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper springer.com

Preface

In 2006 a special semester on Gr¨ obner bases and related methods was organized by RICAM and RISC, directed by Bruno Buchberger and Heinz Engl. The main focus of the semester were the development of the formal theory of Gr¨ obner bases (brieﬂy GB), the eﬃcient implementation of all algorithms related to this theory, and the promotion of recent and new applications of GB. The workshop D1 “Gr¨ obner bases in cryptography, coding theory and algebraic combinatorics”, Linz, May 1–6, 2006 (chairmen M. Klin, L. Perret, M. Sala) was one of the main ingredients of the semester. The last two days of this workshop, devoted to combinatorics, made it possible to bring together experts in algorithmic problems related to coherent conﬁgurations and association schemes with a community of people working in the area of GB. Each side was interested in understanding the computational problems and current algorithmic possibilities of the other, with a particular objective of introducing the practical use of GB in algebraic combinatorics. Materials (mainly slides of lectures and posters) available from the site http://www.ricam.oeaw.ac.at/specsem/srs/groeb/schedule D1.html provide a helpful and vivid picture of the successful exchange of scientiﬁc information during the workshop D1. As a follow-up to the special semester, 10 volumes of proceedings are being published by diﬀerent publishers. The current collection of papers reﬂects diverse investigations in the area of algebraic combinatorics (with or without explicit use of GB), but with a deﬁnite emphasis on algorithmic approaches. Ever since its initial inception in 1965 by Buchberger, the algorithmic theory of GB has been computer-oriented, and nowadays this method is available in all major software systems. The theory of GB has fundamental computational signiﬁcance in algebraic geometry, and it is also applied in many other areas, including coding theory, cryptography, statistics and optimization. In the last few decades computational methods have proved to be highly eﬀective in algebraic combinatorics, particularly in the search for new distance regular and strongly regular graphs and partial geometries, the determination of au-

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tomorphism groups of association schemes, and identiﬁcation of graphs. It is now very timely to combine these eﬀorts in the two areas of GB and algebraic combinatorics, with a special emphasis on algorithmic aspects. A common feature of many problems in combinatorics and geometry is the existence of huge search spaces. Elements of these spaces are parametrized by integer vectors, subject to suitable feasibility conditions, usually expressed in terms of certain equations and inequalities. Typical research tasks for a prescribed feasible set of parameters consist of proving the existence of a desired object, classifying all objects up to isomorphism, and investigating the symmetry properties of discovered objects. Techniques developed within GB provide attractive possibilities for all these types of investigation in combinatorics and geometry. This collection aims to promote the interplay between algorithmic aspects of GB and its existing and possible new combinatorial applications, starting with the immediate eﬃcient use of GB and ending with the consideration of algorithmic approaches which are currently based on alternatives to GB. Part A of this book consists of ﬁve tutorials prepared by participants of the workshop D1. Three of these tutorials, written by members of the same scientiﬁc group, introduce the reader to the methodology of computeraided investigation of coherent conﬁgurations, fusion association schemes and related combinatorial structures, with a special emphasis on the use of the computer algebra system GAP. Sections 1 and 2 of the tutorial T2 by Klin, Reichard and Woldar may serve as a brief introduction to the whole scope of those concepts from algebraic combinatorics which are considered in many other contributions in this book. The main body of T2 describes a new class of Siamese combinatorial objects which appear through the interplay of color graphs, association schemes, Steiner designs and generalized quadrangles. Tutorial T1 by Heinze and Klin aims to present natural links between diverse objects from algebra, geometry and graph theory such as loops, Latin squares, nets, association schemes, strongly regular graphs and partial diﬀerence sets. Tutorial T4 by Pech and Reichard deals with the problem of enumeration of orbits of a permutation group acting on subsets. It describes an eﬃcient implementation of an algorithm for the solution of this problem, based on a depth-ﬁrst approach using the techniques of dynamic programming. Tutorial T3 by Leonard introduces the reader to innovative techniques applying GB in the area of association schemes. We are pleased to mention that this approach was conceived at Linz, in the course of the workshop D1. Together with the above-mentioned texts, this paper provides a vivid picture of computational and theoretic aspects of the theory of association schemes. Special themes of the tutorial T3 are ﬂag algebras of generalized quadrangles and Steiner designs, their ﬁnitely-presented form, and the explicit enumeration of all fusion association schemes via the use of the Buchberger algorithm. Codes in MAGMA, as well as some computational results, greatly strengthen

Preface

vii

the role of T3 as a striking pattern for future new applications of the suggested techniques. Tutorial T5 by Peretz is concerned with the Jacobian conjecture, a famous and ambitious problem which has remained open since 1939. Posed originally in the language of classical analysis and algebraic geometry, it nowadays has very deep links with algebra. A computational approach to the 2-dimensional version of this conjecture based on the use of GB is presented in the framework of diverse reformulations of the problem and its interesting links with recent activities of many mathematicians. It brings the reader to the forefront of innovative attempts to understand the origins of the diﬃculty of this conjecture. The individual tutorials, and indeed the whole of Part A, can play a signiﬁcant training role for researchers: they provide a relatively self-contained introduction to several ﬂourishing areas of modern combinatorics in the framework of a constructive algorithmic approach, with emphasis on the practical use of modern software. Moreover each tutorial contains new scientiﬁc results, generally obtained via the use of the algorithmic tools described here. Part B consists of seven research papers, many of which also serve an instructive purpose by maintaining the ingredients of a tutorial style. Paper R1 by Felszeghy and R´onyai deals with several problems from algebra and combinatorics, which are formulated in terms of a set of points in an n-dimensional vector space over a ﬁeld F . Using techniques of GB and the concept of a Hilbert function of an F -algebra, the authors present the Lex Game method and a few of its striking applications, such as new proofs of Garsia’s generalization of the fundamental theorem on symmetric polynomials, and of the famous rank formula of R. M. Wilson, and an investigation of set families which do not shatter large sets. Paper R6 by Moorhouse also covers a wide range of problems, applying methods of algebraic geometry to diverse problems in ﬁnite geometry. Techniques of GB appear here in a hidden form, for instance via a computational example using the software Macaulay 2. The study of p-ranks of incidence matrices of geometric structures creates a nice interplay with the methodology of paper R1. Paper R4 by Kohnert is devoted to innovative computational eﬀorts in the overlap between coding theory and ﬁnite geometry. Starting with the classical correspondence between two-weight codes and sets of type (d1 , d2 ) in projective spaces, the projective Hjelmslev planes over Galois rings are considered. An extensive computer search leads to new geometric structures and to codes corresponding to them. Techniques of GB may also be applied in the area of mathematical chemistry – this is the main message communicated in R2 by Gugisch. Following the ideas of A. Dreiding & A. Dress, and of N. S. Zeﬁrov & S. S. Tratch, techniques of oriented matroids are revisited in order to provide adequate tools to model conformations of organic compounds. This requires the eﬃcient enumeration of diverse geometrical objects, in particular partial chirotopes. Corresponding

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algorithmic implementations are described, which are tested on a number of attractive examples. Paper R7 by Ziv-Av describes another possible area for the application of GB. The notion of a total graph coherent conﬁguration is introduced and investigated for two classical inﬁnite series of strongly regular graphs – triangular graphs and lattice square graphs. In both cases all fusion association schemes are completely classiﬁed. The methodology used has an obvious overlap with tutorial T3, providing a challenge to apply GB in future for the solution of wider classes of similar problems. The remaining two papers acquaint the reader with diﬀerent algorithmic approaches to the solution of combinatorial problems. Paper R3 by Jørgensen deals with non-symmetric association schemes with three classes. A list of feasible parameters for such primitive schemes of order ≤ 100 is calculated. It contains 24 parameter sets, three of which are eliminated in R3 with the aid of computer search techniques. There remain ten open cases, providing a striking challenge for future researchers. Imprimitive schemes are also carefully investigated, and new theoretical results are presented in conjunction with the discovery of four examples on 36 points which are equivalent to ‘skew’ Bush-type Hadamard matrices of order 36 (examples of such matrices were not known before). Finally, paper R5 by Miyamoto introduces a computational approach to doubly transitive permutation groups via the concept of a superscheme. This approach is investigated for a particular case: 3-designs on q+2 points are constructed, which are invariant with respect to the action of the group PSL(2, q) on the projective line, extended by an extra point. The resulting designs are studied for some small values of q with the aid of the computer package GAP. We are pleased to thank Bruno Buchberger and Peter Paule for their essential support of this project at all stages of its development. The kind patience and attention to detail of Ruth Allewelt (Springer) were crucially helpful. Last but not least we wish to thank the two dozen referees who helped us to ensure hopefully high scientiﬁc standards for the entire collection. Mikhail Klin Gareth A. Jones Aleksandar Juriˇsi´c Mikhail Muzychuk Ilia Ponomarenko March 2009

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Part A – Tutorials T1 Aiso Heinze, Mikhail Klin Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics . . . . . . . . . . . . . . 3 T2 Mikhail Klin, Sven Reichard, Andrew Woldar Siamese Combinatorial Objects via Computer Algebra Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 T3 Douglas A. Leonard Using Gr¨ obner Bases to Investigate Flag Algebras and Association Scheme Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 T4 Christian Pech, Sven Reichard Enumerating Set Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 T5 Ronen Peretz The 2-Dimesional Jacobian Conjecture: A Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Part B – Research papers R1 B´ alint Felszeghy, Lajos R´ onyai Some Meeting Points of Gr¨obner Bases and Combinatorics . . . . . . . . 207 R2 Ralf Gugisch A Construction of Isomorphism Classes of Oriented Matroids . . . . . 229 R3 Leif K. Jørgensen Algorithmic Approach to Non-symmetric 3-class Association Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 R4 Axel Kohnert Sets of Type (d1 , d2 ) in Projective Hjelmslev Planes over Galois Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 R5 Izumi Miyamoto A Construction of Designs from PSL(2, q) and PGL(2, q), q = 1 mod 6, on q + 2 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 R6 G. Eric Moorhouse Approaching Some Problems in Finite Geometry Through Algebraic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .285 R7 Matan Ziv-Av Computer Aided Investigation of Total Graph Coherent Conﬁgurations for Two Inﬁnite Families of Classical Strongly Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Contributors

B´ alint Felszeghy Computer and Automation Institute, Hungarian Academy of Science, Budapest, Hungary; Institute of Mathematics, Budapest University of Technology and Economics, Budapest, Hungary, [email protected] Ralf Gugisch Mathematisches Institut, University of Bayreuth, 95440 Bayreuth, Germany, [email protected] Aiso Heinze Department of Mathematics, Leibniz Institute for Science Education, Olshausenstraße 62, 24098 Kiel, Germany, [email protected] Leif K. Jørgensen Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7, 9220 Aalborg, Denmark, [email protected] Mikhail Klin Ben-Gurion University of the Negev, Beer Sheva 84105, Israel, [email protected] Axel Kohnert Mathematisches Institut, University of Bayreuth, 95440 Bayreuth, Germany, [email protected] Douglas A. Leonard Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA, [email protected]

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Izumi Miyamoto Department of Computer Science and Media Engineering, University of Yamanashi, Kofu 400-8511, Japan, [email protected] G. Eric Moorhouse Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA, [email protected] Christian Pech Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel, [email protected] Ronen Peretz Department of Mathematics, Ben-Gurion University, Beer Sheva, Israel, [email protected] Lajos R´ onyai Computer and Automation Institute, Hungarian Academy of Science, Budapest, Hungary; Institute of Mathematics, Budapest University of Technology and Economics, Budapest, Hungary, [email protected] Sven Reichard University of Western Australia, Crawley 6009, Western Australia, [email protected] Andrew Woldar Villanova University, Villanova, PA 19085, USA, [email protected] Matan Ziv-Av Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel, [email protected]

Part A

Tutorials

Loops, Latin Squares and Strongly Regular Graphs: An Algorithmic Approach via Algebraic Combinatorics Aiso Heinze1 and Mikhail Klin2 1

2

Department of Mathematics, Leibniz Institute for Science Education, Olshausenstraße 62, 24098 Kiel, Germany. [email protected] Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel. [email protected]

Summary. Using in conjunction computer packages GAP and COCO we establish an eﬃcient algorithmic approach for the investigation of automorphism groups of geometric Latin square graphs. With the aid of this approach an inﬁnite series of proper loops is presented which have a sharply transitive group of collineations. The interest in such loops was expressed by A. Barlotti and K. Strambach.

Key words: Latin square graph, Loop, Net, Transversal design, Regular subgroup, Computer algebra, Partial diﬀerence set, Association scheme

1 Introduction The goal of this tutorial paper is to introduce the reader to links between loops, Latin squares, nets, association schemes, strongly regular graphs and partial diﬀerence sets via an algorithmic approach based on the use of computer algebra packages. Simultaneously we are pursuing a serious scientiﬁc objective, introducing an inﬁnite series of proper loops Q2p of order 2p, p a prime, p ≡ 3 (mod 4), for which the group G =Aut(Γ ) contains a regular subgroup of order 4p2 . Here Γ = SRG(Q) is the Latin square graph naturally associated with the loop Q. The problem of the existence of such loops goes back to A. Barlotti and K. Strambach [14]. Originally, we ﬁrst found Q6 , examining Edward Spence’s catalogue [95] of strongly regular graphs that have a relatively small number of vertices. By creating a computer free description of all necessary features of Q6 and related combinatorial structures, we came to the conclusion that in fact Q6 is just the ﬁrst member of an inﬁnite series. The methodology presented by us is based (for every value of p) on a careful inspection of a suitable auxiliary structure say S. We describe the complete M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 1,

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automorphism group of S, which turns out to be isomorphic to the desired group G. After that, starting from the structure S, we justify all requested properties of the graph Γ and loop Q. The topics considered here are the subject of various investigations in diverse areas of mathematics. According to the experience accumulated by us, in some extent these areas still remain isolated. A researcher acting in one area may not even be aware of the existence of a “parallel world” in another area. A genetical style of exposition, utilized in this paper, is designed to help the reader to overcome quickly artiﬁcial terminological barriers and to exploit all advantages resulted from simultaneous operation with adequate algebraic, geometric and combinatorial objects. The article consists of ten sections. In Sect. 2 the most important preliminaries are introduced in order to make this text self-contained as much as it is possible. Classical and folklore results about the links between the considered structures are presented in Sect. 3. These and some other links are illustrated with the aid of examples in Sect. 4. In Sects. 5 and 6 we explain the origins of our interest and discuss brieﬂy how various computer tools were used on the initial stages of the project. Section 7 is the main part of the article. In fact, by examining Q6 , we are able to illustrate all of the important properties of the members of the whole inﬁnite series. The next Sect. 8 serves as a bridge from Q6 to the general case considered in Sect. 9. Section 10 contains miscellaneous information which was postponed to the end of the presentation in order to not interrupt the main, we believe quite friendly, line of exposition. In this section, in particular, an alternative approach is outlined which makes it possible to generalize our results for all prime values of the parameter p. We are trying to provide also all important credits, in particular to the crucial inputs of A. Sprague, R. Wilson and K. Kunen. Our comprehensive bibliography may serve as a reasonable complement to the one included in [69].

2 Preliminaries 2.1 Main Notions 2.1.1 We start from a classical deﬁnition of a Latin square (see e.g., [65]). A Latin square of order n is a quadruple (R, C, S; L) where R, C, S are sets of cardinality n, called rows, columns and symbols respectively and L is a mapping L : R × C → S such that for any i ∈ R and any x ∈ S the equation L(i, j) = x has a unique solution j ∈ C, and for any j ∈ C, x ∈ S the same equation has a unique solution i ∈ R. In a more naive way the above rigorous deﬁnition may be interpreted as an n × n array with n diﬀerent entries, n ≥ 2, such that each entry occurs exactly once in any row and in

Loops, Latin Squares and Strongly Regular Graphs

5

any column of the array. As a rule, we will set R = C = S = [1, n], where [1, n] = {x ∈ N |1 ≤ x ≤ n} for n ∈ N . A Latin square is said to be reduced or to be in standard form, if in the ﬁrst row and column its elements 1, 2, . . . , n occur in natural order. A quasigroup is a set Q with a binary operation “·” such that for all a, b ∈ Q the equations a · x = b and y · a = b have a unique solution in Q. It is easy to see that every Latin square may be interpreted as a multiplication table of a quasigroup, and for each quasigroup its Cayley table provides a Latin square. A loop L is a quasigroup with an identity element e ∈ L with the property ex = xe = x for every x ∈ L. Usually, the identity element e is identiﬁed with 1 ∈ [1, n]. Then we may say that each loop naturally deﬁnes a reduced Latin square and each reduced Latin square may be interpreted as a Cayley table of a loop. An associative loop is a group. We refer to [34] and [79] as to classical sources in the theory of Latin squares and quasigroups respectively. 2.1.2 There are a few alternative representations of Latin squares, as well as some related combinatorial structures, which are proved to be very eﬃcient in the classiﬁcation of Latin squares. Let V be a set of cardinality n and let L ⊆ V 3 be a ternary relation over V (that is a collection of ordered triples whose components belong to V ) such that |L| = n2 . Then the relation L is called a Latin square of order n if and only if each of the following three sets has n2 distinct elements: L1 = {(i, j)|(i, j, k) ∈ L}, L3 = {(j, k)|(i, j, k) ∈ L}.

L2 = {(i, k)|(i, j, k) ∈ L},

We refer to [6] for a detailed algorithmic analysis of this triple representation for the goals of a constructive enumeration. In the case when a Latin square is originally represented as the Cayley table of a group H, it is convenient to identify its triple representation with a set of n2 ordered triples {(i, j, k)|i, j, k ∈ H, ijk = 1}, where 1 is the identity element of H. An orthogonal array OA(n, 3) of order n and depth 3 is a 3 × n2 array with entries from [1, n], such that for any two rows of the array the n2 vertical pairs occurring in these rows are diﬀerent. 2.1.3 A 3-net of order n is an incidence structure S = (P, L) which consists of an n2 -element set P of points and a 3n-element set L of lines. The set L is partitioned into three disjoint families L1 , L2 , L3 of (parallel) lines, for which the following conditions hold: (i) every point is incident with exactly one line of each family Li (i = 1, 2, 3); (ii) two lines of diﬀerent families have exactly one point in common; (iii) two lines in the same family do not have a common point;

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(iv) there exist three lines belonging to three diﬀerent families which are not incident with the same point. The families L1 , L2 , L3 sometimes are called the directions or parallel classes of S. It is easy to understand that each Latin square L of order n naturally produces a 3-net. Points of this net are formed by the cells of L, while its directions correspond to horizontal lines, vertical lines and the lines occupied in L by the same element. Thus, a 3-net S is a uniform and regular incidence structure with the parameters v = n2 , b = 3n, k = n, r = 3 (see e.g., [52] for all deﬁnitions). Note also that a 3-net S is a particular case of a partial geometry (see e.g., [65]). It is also clear that we can get a Latin square from a 3-net, identifying one direction with rows, a second with columns and the last one with the symbols. Note that a 3-net is a particular case of a k-net (k ≥ 3). In what follows we will call a 3-net simply a net, provided that there is no risk of a confusion. 2.1.4 Now we may consider a dual structure S T = (L, P) which has L as points and P as lines, and with the incidence relation transposed. (Recall that in our case the original incidence relation coincides with the set-theoretical inclusion.) Then S T has three families of points each of cardinality n, which are called groups3 , and n2 blocks (lines). The parameters of S T are v = 3n, b = n2 , k = 3, r = n. By deﬁnition two distinct points from the same group are not collinear, while there is exactly one line through two distinct points from distinct groups. A structure S T will be called a transversal design T D(3, n). Note that a transversal design is a particular case of a partial linear space (or conﬁguration in other terms), see e.g., [21]. 2.1.5 The reader is referred to [13, 21, 40] for various aspects of the theory of association schemes. Our presentation here follows mainly [45]. Let L be a Latin square represented in a traditional manner as an n × n array and let Ω be the n2 -element set of cells of L. Let R0 = {(x, x)|x ∈ Ω} be the complete diagonal relation over Ω. Deﬁne R1 as follows: (x, y) ∈ R1 for x, y ∈ Ω if and only if x, y are in the same row of L. Similarly R2 is deﬁned with respect to the columns of L, while for two cells x, y we have (x, y) ∈ R3 if and only if x, y are occupied in L by the same symbol. Finally, let R4 = Ω 2 \ {R0 ∪ R1 ∪ R2 ∪ R3 }. Then it is easy to see, that the relational structure M := M(L) = (Ω, {R0 , R1 , R2 , R3 , R4 }) is an association scheme with four classes. (This is a particular case of amorphic association schemes introduced in [45].) Merging of any classes in M provides again an association scheme. In particular, (Ω, {R0 , R1 ∪R2 , R3 , R4 }) is 3

Note that the term “group” in the deﬁnition of T D(3, n) is very unfortunate causing confusion with the term “group” in algebra.

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7

an association scheme of L with three classes, and (Ω, {R0 , R1 ∪R2 ∪R3 , R4 }) is an association scheme with two classes. Each of its classes corresponds to a strongly regular graph. A strongly regular graph Γ = SRG(L) = (Ω, R1 ∪R2 ∪R3 ) is commonly called a Latin square graph, it has parameters v = n2 , k = 3(n − 1), λ = n, μ = 6. 2.1.6 Two permutations are discordant if no symbol has the same image under both permutations. A set of n pairwise discordant permutations of degree n is called a complete set of discordant permutations (an alternative name is a sharply transitive set of permutations). There are three important ways to establish a one-to-one correspondence between Latin squares of order n and complete sets of discordant permutations of degree n. For the ﬁrst correspondence we regard permutations as various rows of a Latin square L. The second correspondence (dual to the ﬁrst) is linked to columns of L. Usually this correspondence is not distinguished from the ﬁrst one. Another correspondence attributes to each permutation a permutation matrix, thus showing positions in L where a certain element i appears. In our consideration these correspondences will sometimes be viewed implicitly. We refer to [35, 36] for a detailed consideration of these ways. Very close to permutation representations are graphical representations of Latin squares, using certain factorizations of the complete n-vertex undirected graph Kn , complete directed graph Kn∗ and complete directed pseudograph ∗ K n with a loop in each vertex. 2.2 Classiﬁcation Here we brieﬂy discuss the most signiﬁcant ways to classify Latin squares. 2.2.1 Let Ln denote the total number of Latin squares of order n considered as n × n-arrays over [1, n]. Let Rn denote the total number of reduced Latin squares of order n. It is easy to see that Ln = n!(n − 1)!Rn . 2.2.2 In what follows we denote by Ln a set of all Latin squares of order n in their ordered triple representation, i.e., for L ∈ Ln the triple, (i, j, k) ∈ L means that the cell on the intersection of row i and column j is ﬁlled by the symbol k. Thus, there exists a natural bijection between the sets of triple representations and the traditional n × n-array representations. We denote by Sn a symmetric group of degree n and order n! in its natural action on [1, n]. Following [6], let us denote by Gn the exponentiation Sn ↑ S3 of the symmetric group of degree n with the symmetric group of degree 3. (Sometimes the notation [Sn ]S3 is used instead of Sn ↑ S3 .) Recall that Sn ↑ S3 = {[g, h(x)]|g ∈ S3 , h(x) ∈ Sn for x ∈ [1, 3]} consists of formal tables [g, h(x)] = [g; h1 , h2 , h3 ], where g ∈ S3 , h1 , h2 , h3 ∈ Sn . In order to get the

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image (α1 , α2 , α3 )[g,h(x)] for a triple (α1 , α2 , α3 ) ∈ [1, n]3 we ﬁrst apply hi to αi , i = 1, 2, 3, and after that permute the components of the resulted triple according to the permutation g. Thus, Sn ↑ S3 is a permutation group of degree n3 and of order 6(n!)3 . This group acts transitively on the set [1, n]3 . As an abstract group the group Sn ↑ S3 is isomorphic to the wreath product of the groups Sn and S3 acting on a 3n-element set. We refer to [59] and [40] for a more detailed discussion of the operation of the exponentiation and its combinatorial applications. Proposition 1. The group Gn = Sn ↑ S3 acts on the set Ln . 2.2.3 Let (Q1 , ◦) and (Q2 , ) be two quasigroups. We introduce an isomorphism of quasigroups in a traditional algebraic manner, that is, if there exists a bijection f from Q1 to Q2 such that for a, b, c ∈ Q1 we have (a ◦ b = c)

⇐⇒

(af bf = cf ).

The set Aut(Q) of all isomorphisms of Q onto itself (in other words the set of automorphisms of Q) evidently forms a group, which can be regarded as a permutation group of degree n = |Q|. Establishing isomorphism classes of quasigroups is an important ingredient in the classiﬁcation of Latin squares. A similar deﬁnition and problem may be formulated for loops. Here of course, any isomorphism of loops L1 and L2 should send the identity of L1 to the identity of L2 . 2.2.4 Let (Sn )3 = Sn ↑ E3 be a subgroup of Sn ↑ S3 . Here E3 is the trivial subgroup in S3 , which is generated by the identity e ∈ S3 . In other words, (Sn )3 = {[e; h1 , h2 , h3 ]|h1 , h2 , h3 ∈ Sn }. Two Latin squares L1 , L2 ∈ Ln are called isotopic, if they belong to the [e;h ,h ,h ] same orbit under the action of (Sn )3 on Ln , e.g., L2 = L1 1 2 3 . Intuitively, we may explain that two Latin squares L1 , L2 represented as n × n-arrays are isotopic, if we can ﬁnd three independent permutations h1 , h2 , h3 ∈ Sn , such that the action of h1 on rows, h2 on columns, h3 on symbols of L1 brings L1 to L2 . The orbits (equivalence classes) of this relation are called isotopy classes of Latin squares. The stabilizer Is (L) = {σ ∈ (Sn )3 |Lσ = L} is called the autotopy group of L, the elements of Is (L) are autotopisms of L. Note, that in these terms isomorphism (automorphism) of a quasigroup (loop) may be formulated as a particular case of isotopy. Namely, σ = [e; h1 , h2 , h3 ] ∈ (Sn )3 is an isomorphism of L1 and L2 if and only if it is an isotopism between L1 and L2 , for which h1 = h2 = h3 . 2.2.5 Let us now pay special attention to the group S3 which participated in the deﬁnition of Gn = Sn ↑ S3 . This group S3 is acting on the set [1, 3] = {1, 2, 3}. Let t = (1, 2)(3) ∈ S3 be one of the transpositions from S3 .

Loops, Latin Squares and Strongly Regular Graphs

9

Then t is evidently a subgroup of order 2 in S3 and we may also consider a permutation group Sn ↑ t of order 2 · (n!)3 , where Sn ↑ t = {[g; h1 , h2 , h3 ]|g ∈ t, h1 , h2 , h3 ∈ Sn }. We say that the Latin squares L1 and L2 are of the same type if L2 = Lσ1 for σ ∈ Sn ↑ t. Type equivalence can be explained intuitively in terms of n × n-arrays, if in addition to isotopisms we allow the operation of the transposition of arrays. 2.2.6 Let us consider the orbits of a natural action (Gn , Ln ); they will be called the main classes of Latin squares of order n. Two squares from the same main class will be called paratopic. (Recall that we postpone the discussion of the origins of all the terminology used by us, as well as other alternatives to Sect. 10.) The stabilizer M C(L) = {σ ∈ Gn |Lσ = L} is called the autoparatopy group of L, and its elements are the autoparatopisms of L. 2.2.7 Another traditional equivalence class of Latin squares is related to the consideration of the exponentiation En ↑ S3 , where En is the identity permutation group of degree n, i.e., En ↑ S3 = {[g; e, e, e]|g ∈ S3 , e is the identity in Sn }. Evidently En ↑ S3 is a permutation group of order 6, acting on the set Ln . The orbits of this action are called the conjugate classes of Latin squares of order n. Two Latin squares belonging to the same conjugacy class are called conjugates. We can also introduce a stabilizer of a Latin square in the group En ↑ S3 . It is playing a signiﬁcant role in the quasigroup theory, with its use one may introduce a few important classes of quasigroups. Note that the action of a group En ↑ S3 on Ln has a very natural intuitive interpretation. This means that we allow permutations of the sets of rows, columns and symbols, and conjugate squares are those, which appear one from another via such a permutation. 2.2.8 Let L be a Latin square of order n, let N (L) be the 3-net, deﬁned by L. Then we say that two Latin squares L1 and L2 are net-equivalent if and only if N (L1 ) ∼ = N (L2 ). Here, by an isomorphism of 3-nets, we mean a traditional isomorphism of incidence structures. It is quite evident, that the introduced net-equivalence is coinciding with the AS-equivalence of Latin squares. By AS-equivalence of L1 and L2 we mean that the 4-class (amorphic) association schemes M(L1 ) and M(L2 ) are isomorphic. The automorphism group Aut(M(L)) of the association scheme M = (Ω, {R0 , R1 , R2 , R3 , R4 }) is the traditional automorphism group of the

10

Aiso Heinze and Mikhail Klin

association scheme, that is Aut(M) = {g ∈ S(Ω)|Rig = Ri , 0 ≤ i ≤ 4}, where S(Ω) is the symmetric group of the set Ω. It is also important to consider the group of color (or weak) automorphisms of M, namely CAut(M) = {g ∈ S(Ω)|Rig = Rj , 0 ≤ i, j ≤ 4}. Permutations from CAut(M) may permute classes of M, while permutations from Aut(M) preserve each class of M. These two natural groups also have other names, provided that we consider 3-nets instead of association schemes. The collineation group Σ of a quasigroup Q is the (full) collineation group of the 3-net N (Q). Collineation is deﬁned to be a permutation of points of N (Q), which maps a line onto a line. The group Σ has a normal subgroup T of index ≤ 6, which maps every class of (parallel) lines onto itself. This group may be called the group of direction preserving collineations of N (Q). If it is necessary to attribute T and Σ to Q, we write T (Q) and Σ(Q). 2.2.9 Let L1 , L2 ∈ Ln and let Γ1 = SRG(L1 ), Γ2 = SRG(L2 ) be strongly regular graphs deﬁned by L1 and L2 respectively. We say that L1 and L2 are SRG-equivalent if Γ1 and Γ2 are isomorphic graphs. If L is a Latin square and Γ = SRG(L), then we also consider Aut(Γ ), the classical automorphism group of the graph Γ . This group will play the most signiﬁcant role in this paper, providing a natural graph-theoretical way for measuring the symmetry of L. 2.3 Regular Subgroups In this paper we are especially interested in regular subgroups of the collineation group Σ of a quasigroup (loop). Recall that a ﬁnite permutation group (H, Ω) is called regular, if it is transitive and the order |H| is equal to the degree |Ω|. Each regular group H is similar (as a permutation group) to a right action of H on its elements. Therefore, we always can regard H as acting on itself and thus denote it by (H, H). Following [14], we will call a regular subgroup of Σ(Q), where Q is a quasigroup, a sharply transitive group of collineations of Q. We can also consider a regular subgroup H of the group T (Q), where Q is a quasigroup. It is clear that in this case elements of H preserve each of the three directions of the net N (Q). Following A. Sprague [96] such a subgroup H will be called a translation group of N (Q), while a net N (Q), for which a translation group exists, will be called a translation net. In Sect. 3 we will show that if Q is a group, then Σ(Q) contains a sharply transitive collineation group H, moreover H is also a translation group. A loop Q will be called a proper loop if its main class does not contain a group. This paper was originally motivated by the following remark by Barlotti and Strambach, see [14, p. 79]: “We were not able to decide whether there exists a proper ﬁnite loop having a sharply point transitive group of collineations.”

Loops, Latin Squares and Strongly Regular Graphs

11

A question about the existence of sharply transitive collineation groups can be reformulated in terms of graphs. Let H be a group (in additive notation) and let X be a subset of H. Then a (directed) graph Γ = Cay(H, X) = (H, R), where R = {(h, x + h)|h ∈ H, x ∈ X} is called a Cayley graph over H with a connection set X. The graph Γ does not have loops, if the identity element 0 ∈ H does not belong to X. We can identify Γ with a simple (undirected) graph, if −X = X, where −X = {−x|x ∈ X}. Finally, if Γ = Cay(H, X) is a strongly regular Cayley graph (over a group H) then its connection set X is called a partial diﬀerence set in H. The investigation of the partial diﬀerence sets in groups of small order in [50] was, in fact, the starting point of this project. 2.4 Principal Loop-Isotopes of Quasigroups There is one more extremely important class of groups, which may be associated to a given quasigroup Q. We start from a subgroup P of Gn , which consists of tables [e; g1 , g2 , ], where e is the identity of S3 , g1 , g2 ∈ S(Q) and is the identity of S(Q). This subgroup P has order (n!)2 . Two quasigroups are called principally isotopic, if their isotopism can be realized by an element θ ∈ P. Those loops L, which are principally isotopic to a quasigroup Q, are called its principal L-isotopes. For any a, b ∈ Q we can deﬁne in a sense a canonical principal L-isotope of Q, which is denoted by L(a, b), see [26] for a precise deﬁnition. In terms of these n2 objects R. Bryant and H. Schneider deﬁne in [26] a certain group, which is associated with a given quasigroup Q. Following D. A. Robinson [82], we will call this group the Bryant-Schneider group of Q, or brieﬂy BS(Q). It turns out that the Bryant-Schneider group plays a signiﬁcant role in the enumeration of loops which are isotopic to Q.

3 Classics and Folklore In this section we collect the most important facts about the links between Latin squares and various kinds of algebraic and combinatorial objects. We will give only brief formulations without any attempts to the consideration of proofs. Some of the results are of a deﬁnite folklore nature. The most important references will be mentioned immediately, the other ones will be discussed in the last section of the article. However, a complete characterization of all bibliographical sources is beyond the scope of this paper. Some part of the material here is also considered in [32]. 3.1 General Links Between Groups Let L be a Latin square of order n, and let N (L) be a 3-net which is deﬁned by L. Let M = M(L) be an association scheme with four classes, naturally

12

Aiso Heinze and Mikhail Klin

attributed to L. Three of the classes of M deﬁne imprimitive strongly regular graphs, each isomorphic to n ◦ Kn . Each such graph consists of n copies of the complete graph on n vertices and thus, corresponds to one parallel class of the net N (L). Conversely, if we have a net N (L), we easily determine an association scheme with four classes, which is denoted here as M(L). From these observations it is easy to get that Aut(M(L)) = T (L) as it was deﬁned in Sect. 2.2.8. One may also consider a group CAut(M(L)) of weak (or color) automorphisms of the association scheme M(L). A combinatorial deﬁnition of this group was also given in Sect. 2.2.8. We see that the group CAut(M(L)) coincides with the group Σ(L) of all collineations of L. Let us now start to discuss the link between Aut(N (L)) and Aut(SRG(L)), where Γ = SRG(L) is a strongly regular graph deﬁned by L. Here we have the particular case of a more general situation. Namely, N (L) is a special case of a partial geometry, while Γ is the point graph of this geometry. Clearly Γ is deﬁned by N (L), however N (L) is not necessarily uniquely reconstructed from Γ . For more details we refer to the classical papers [22, 24, 20] and to textbook [65]. Therefore, in general we may only say that Aut(N (L)) ≤ Aut(SRG(L)), and moreover, it may happen that the ﬁrst group is a proper subgroup of the second. 3.2 Geometrical and Nongeometrical SRG’s Let Γ be a strongly regular graph with the parameters v = n2 , k = 3(n − 1), λ = n, μ = 6. Such a graph is usually called a pseudo-Latin square graph. The name refers to the fact that for an arbitrary Latin square L of order n the graph SRG(L) is indeed a strongly regular graph with the same parameters. In this case SRG(L) is called simply a Latin square graph. Note again that a similar situation appears for an arbitrary partial geometry with the parameters K, R, T . In our case we have K = n, R = 3, T = 2. The question, when a pseudo-geometrical strongly regular graph is indeed a geometrical graph, was considered by Bruck and Bose in [22, 24, 20]. We formulate their general answer for a particular case of 3-nets. Proposition 2. A strongly regular graph Γ with the parameters v = n2 , k = 3(n − 1), λ = n, μ = 6 is a Latin square graph, provided that n > 23 (see extra comments in Sect. 10). We will discuss the small cases n = 3, 4, 5 in the next section. 3.3 Factorization of Latin Square Graphs Now we are interested in the following question. Suppose that Γ is a SRG(L) for a Latin square graph L of order n. Then the edge set of Γ is attained via the merging of edge sets of three copies of graphs n ◦ Kn , which correspond

Loops, Latin Squares and Strongly Regular Graphs

13

to the three parallel classes of the net N (L). Therefore, there exists a special factorization of the graph Γ . In what follows for a Latin square graph Γ we will call an arbitrary collection of n disjoint cliques of size n in Γ a spread of Γ . Then a 3-net is nothing but a factorization of Γ into three spreads. In general, such a factorization is not unique (see examples in Sect. 4). The following lemma is also of a folklore nature. Lemma 1. Let L be a Latin square and Γ = SRG(L). If n ≥ 5, then the cliques of Γ necessarily correspond to lines of an associated 3-net N (L). Lemma 2 (cf. [7]). For n ≥ 5 we can reconstruct the 3-net N (L) uniquely from the graph Γ = SRG(L). This lemma immediately implies the following important graph theoretical reformulation. Proposition 3. For n ≥ 5 we have Aut(SRG(L)) = Aut(N (L)) = Σ(L). 3.4 The Group Case In this section we pay special attention to a case when a Latin square L is a Cayley table of a group H. Such a case of a Latin square will be called a group case. Sometimes we may write L = L(H) to stress that L is coming from a group H. Lemma 3 (cf. [32]). Let H be a ﬁnite group and L := {(x, y, z) ∈ H 3 | xyz = 1} an associated Latin square considered in a special triple representation. Deﬁne mappings L → L by (a) (b) (c) (d)

ι : (x, y, z) → (z −1 , y −1 , x−1 ), σ : (x, y, z) → (y, z, x), τa,b,c : (x, y, z) → (a−1 xb, b−1 yc, c−1 za), where (a, b, c) ∈ H 3 , αf : (x, y, z) → (xf , y f , z f ) where f ∈ Aut(H).

Then each of the mappings above is an automorphism of L and Aut(L) = ι, σ, τa,b,c , αf (a,b,c)∈H 3 ,f ∈Aut(H) . Corollary 1. Let L = L(H) be a group Latin square. Then Aut(N (L)) ∼ = (H 2 : Aut(H)).S3 . Theorem 1 (cf. [32]). Let L(H) be a group Latin square with |H| ≥ 5. Then Aut(SRG(H)) ∼ = (H 2 : Aut(H)).S3 . Clearly, the proof is based on the combination of Proposition 3 with Corollary 1. In the next section we will discuss some special examples. Corollary 2. Let H be a group of order n and Γ = SRG(H). Then (a) Aut(Γ ) is a transitive group of degree n2 , (b) Aut(Γ ) contains a regular subgroup H 2 of order (and degree) n2 .

14

Aiso Heinze and Mikhail Klin

3.5 Main Class of a Group Case We refer to [72], see also [44], Exercise 10.23 as strict formulations of the following facts of folklore nature. Proposition 4. Let H be a group of order n and let Q be a loop of order n. Then H ∼ = Q if and only if the corresponding 3-nets N (H) and N (Q) are isomorphic. Corollary 3. (a) If H1 and H2 are nonisomorphic groups of order n, then SRG(H2 ). SRG(H1 ) ∼ = (b) If a Latin square L does not appear in a main class of any group, then SRG(L) is not isomorphic to any Latin square graph over a group. This culminating corollary will play an essential role in all further considerations.

4 Garden of Small Examples 4.1 Main Results The goal of this section is to provide the reader an opportunity to digest many of the introduced notions on a level of simple small examples. We restrict ourselves to a consideration of examples of order n ≤ 6. Table 1 provides general information about the size of data which is relevant to our consideration. Clearly, n = 1 is an absolutely trivial case. Thus, we start to consider our examples from n = 2. Though this case still should be regarded as trivial, it is quite convenient on this level to illustrate some concepts. Table 1. Data for n ≤ 6 n 1 2 3 4 5 6

Main classes 1 1 1 2 2 12

Types 1 1 1 2 2 17

Isotopy classes 1 1 1 2 2 22

Loops 1 1 1 2 6 109

Reduced squares Rn 1 1 1 4 56 9408

Quasigroups 1 1 5 35 1411 1130531

Total amount of squares |Ln | 1 2 12 576 161280 812851200

Loops, Latin Squares and Strongly Regular Graphs

15

4.2 The Case n = 2 It is easy to check, that |L2 | = 2. The two Latin squares of order 2 are depicted below. 1 2 2 1 2 1 1 2 The group G2 has order 3! · (2!)3 = 48. If we regard G2 as the exponentiation S2 ↑ S3 then the natural action of the latter group is on the set [1, 2]3 of all binary sequences of length 3 (here it is more convenient for us to substitute the alphabet {0, 1} by {1, 2}). In this notation G2 is nothing else but the automorphism group of a very classical object: the 3-dimensional cube Q3 . Following [13], for this graph we will also use the notation H(3, 2) and call it the Hamming graph (with prescribed parameters). Its vertices are all binary strings of length 3, where two vertices are adjacent if and only if corresponding strings diﬀer in exactly one position. Let us now consider cocliques of H(3, 2) of the size 22 = 4, i.e., induced subgraphs of Q3 which are isomorphic to the 4-vertex empty graph E4 . Clearly, Q3 contains just two such cocliques, which are depicted in Fig. 1. The reader will easily attribute these cocliques to the above mentioned Latin squares of order 2. Indeed, we interpret a string (α, β, γ) as a cell of the square on the intersection of a row α and a column β, which is ﬁlled with the element γ. Clearly, the group G2 has just one orbit on such cocliques. This justiﬁes (an evident) fact, that the number of main classes for n = 2 is equal to 1. 4.3 The Case n = 3 Here |G3 | = (3!)4 = 1296 and |[1, 3]3 | = 27. Thus S3 ↑ S3 acts naturally on the 27-element set. Again we regard G3 as the automorphism group of the suitable Hamming graph H(3, 3). This is a distance regular graph of diameter 3 and valency 6. Easy reasonings show that there are exactly 12 diﬀerent Latin squares of order 3, which are presented in Table 2. Again we would like to interpret each Latin square as a certain subgraph of H(3, 3). Starting from a deﬁnition of a Latin square as an array of order 3, we get its interpretation as a coclique of H(3, 3) of size 9. A prescribed vertex of H(3, 3), say (1, 1, 1), appears in exactly 12·9 27 = 4 such cocliques. One of these

Fig. 1. Cocliques of H(3, 2)

16

Aiso Heinze and Mikhail Klin Table 2. All Latin squares of order 3

1

2

3

1

2

3

1

3

2

1

3

2

2

3

1

3

1

2

2

1

3

3

2

1

3

1

2

2

3

1

3

2

1

2

1

3

Z3,1

LU1

RU1

I

2

1

3

2

1

3

2

3

1

2

3

1

1

3

2

3

2

1

1

2

3

3

1

2

3

2

1

1

3

2

3

1

2

1

2

3

SQ1

RU2

Z3,2

LU2

3

1

2

3

1

2

3

2

1

3

2

1

1

2

3

2

3

1

1

3

2

2

1

3

2

3

1

1

2

3

2

1

3

1

3

2

Z3,3

LU3

RU3

SQ2

cocliques is depicted in the diagram of H(3, 3) in Fig. 2, it corresponds to the Latin square Z3,1 in Table 2. (Each line in the picture designates a clique of size 3 in H(3, 3).) Considering the orbits of this coclique with respect to suitable subgroups of G3 , we may ﬁnd a conﬁrmation of the ﬁrst four entries in the row for n = 3 in Table 1. The only non-trivial result, which is relevant to n = 3, is concerned with the number of isomorphism classes of quasigroups. For this purpose, we have to make visible to the reader the action of the diagonal subgroup D3 = {[e; g, g, g]|g ∈ S3 } of order 6 in G3 . An easy task is to describe orbits of D3 on [1, 3]3 . There are ﬁve such orbits (αβγ is a short notation for (α, β, γ)): A = {111, 222, 333}, B = {112, 113, 221, 223, 331, 332}, D = {122, 133, 211, 233, 311, 322},

C = {121, 131, 212, 232, 313, 323}, E = {123, 132, 213, 231, 312, 321}.

All twelve Latin squares of order 3 are presented in Table 2, together with their names, which will be clariﬁed later. Now we are taking ﬁve speciﬁc Latin squares from the table and count for them a number of numerical characteristics. Namely, for each of the ﬁve orbits of D3 we ﬁnd the size of the intersection of it with a subset of [1, 3]3

Loops, Latin Squares and Strongly Regular Graphs

17

Fig. 2. Hamming graph H(3, 3)

corresponding to a square. Besides this, for two invariant subsets of D3 we describe the isomorphism type of a colored subgraph of H(3, 3), induced by the square on the subset (here color means the distance in H(3, 3), the sign [x] shows the induced subgraph). All these data are presented in Table 3. The reader is able to check that the selected squares may be distinguished with the aid of the columns A, [A ∪ B] and [C]. This means that we have at least ﬁve diﬀerent orbits of quasigroups of order 3. Finally, a simple routine inspection shows that we indeed have exactly ﬁve such orbits. Information about these isomorphism classes together with their names (traditionally attributed in quasigroup theory) is presented in Table 4. We believe that the case n = 3 provides a nice illustration of the opportunities Table 3. Characteristics for ﬁve Latin squares Square

A

B

C

D

E

Z3,1

1

2

2

2

2

I

3

0

0

0

6

LU1

1

2

2

2

2

RU1

1

2

2

2

2

SQ1

0

3

3

3

0

[A ∪ B]

[C]

18

Aiso Heinze and Mikhail Klin Table 4. Isomorphism classes of Latin squares of order 3

Name Amount |Aut(Q)|

Cyclic group of order 3 3 2

Idempotent quasigroup 1 6

Left unit quasigroup 3 2

Right unit quasigroup 3 2

Commutative quasigroup 2 3

of the use of Hamming graphs H(n, q) for the classiﬁcation of quasigroups of order q, see also the brief comments in Sect. 10. Unfortunately, the direct use of the graphs for q ≥ 4 becomes impractical. Note also that the unique Latin square graph of order 3 and the unique 3-net of order 3 can be easily described, starting from any of the considered Latin squares. 4.4 The Case n = 4 There are exactly four reduced squares which are presented below. 1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

2

1

4

3

2

1

4

3

2

3

4

1

2

4

1

3

3

4

1

2

3

4

2

1

3

4

1

2

3

1

4

2

4

3

2

1

4

3

1

2

4

1

2

3

4

3

2

1

Thus, the total amount of |L4 | is equal to 4! · 3! · 4 = 576. In principle, the classiﬁcation of quasigroups may still be arranged by hand computations in the same spirit as for n = 3. However, the limits of our exposition do not allow to present the necessary computations. Therefore, we leave the resulting number 35 without justiﬁcation. Clearly, for larger values of n this part of the problem requires deﬁnitively the use of a computer. Let us now turn to the left end of the row in Table 1 corresponding to n = 4. There exist two groups of order 4: the cyclic group Z4 and the elementary abelian group E4 . Thus, according to Corollary 3, there exist at least two main classes. To prove that these are all main classes, we may employ a classical theorem by Shrikhande [91], which claims that a strongly regular graph with the parameters v = n2 , k = 2(n − 1), λ = n − 2, μ = 2 is for n = 4 unique up to isomorphism and is isomorphic to the lattice square graph L2 (n). The vertices of this graph are elements of [1, n]2 and two pairs (a, b) and (c, d) are adjacent if and only if they coincide in exactly one coordinate. It is easy to prove that Aut(L2 (n)) ∼ = Sn ↑ S2 is a group of order 2 · (n!)2 . The same theorem claims that for n = 4 there is one more exceptional strongly regular graph, which is called Shrikhande graph and is denoted by Sh. Now we turn to the Latin square graphs for n = 4. Clearly, these graphs have parameters v = 16, k = 9, λ = 4, μ = 6. It turns out that this parameter set corresponds to the complements of the strongly regular graphs

Loops, Latin Squares and Strongly Regular Graphs

19

with parameters of L2 (4). Thus, taking into account that there exist exactly two non-isomorphic groups of order 4, we obtain the desired result: there are exactly two main classes of Latin squares of order 4. Let us ﬁrst associate L2 (4) to one of the two main classes. Consider the graph Γ1 = L2 (4) : Aut(Γ1 ) has order 2 · (4!)2 = 1152 and acts transitively on the vertices. We want to describe all cliques of size 4 in Γ1 . Let us ﬁx one vertex, say (1, 1) and consider the neighbor subgraph of it in Γ1 . Clearly, it is isomorphic to L2 (3). An easy exercise is to prove that L2 (3) ∼ = L2 (3). It is evident that L2 (3) (and thus L2 (3)) has exactly six diﬀerent 3-cliques. = 24 4-cliques in Γ1 . In order to get a Therefore, altogether there are 6·16 4 3-net of order 4 from the graph Γ1 we need just twelve cliques. Note also that the graph L2 (4) itself has exactly eight cliques, which come from two parallel classes. At this point we consider the graph SRG(Z4 ) for the group Z4 , which corresponds to the third Latin square above. Consider SRG(Z4 ) and detect that the neighbor graph of a vertex in it, say (1, 1) is isomorphic to the hexagon C6 . Clearly, C6 does not have 3-cliques, and as a consequence SRG(Z4 ) does not contain 4-cliques. This implies ﬁnally, that L2 (4) is isomorphic to SRG(E4 ), where E4 is represented by the ﬁrst Latin square above. Recall that Aut(E4 ) ∼ = S3 , and using Corollary 1, we get that Aut(N (E4 )) is a group of order 42 · 3! · 3! = 576. In other words Aut(N (E4 )) is a subgroup of index 2 in Aut(SRG(E4 )). We are obtaining a desired example, which shows that the assumption |H| ≥ 5 in Theorem 1 is important: For groups of order 4 this theorem is not correct. Moreover, the two independent portions of information, which are presented above, now converge. Indeed, 24 cliques in L2 (4) provide two distinct 3-nets, which are isomorphic. The point graphs of both nets are identical. An “extra” automorphism of L2 (4) interchanges the two nets. Let us keep the notation Γ1 ∼ = L2 (4) ∼ = SRG(E4 ). Thus, naturally, we consider Γ2 ∼ = SRG(Z4 ). We already know that Γ 2 ∼ = Sh, and Γ 2 is not geometrical, that is, it is not the point graph of a 2-net. We now want to describe the order of the group Aut(Sh). The Shrikhande graph is a Cayley graph over Z24 with a connection set X = {11, 12, 21, 23, 32, 33} (here Z4 = {0, 1, 2, 3}). We invite the reader to check that Sh can be depicted with the aid of Fig. 3 (which should be considered as drawn on the torus). The group Aut(Sh) also acts transitively on the vertices of Sh. An easy inspection shows that the stabilizer of a point acts on its neighbor set as the dihedral group D6 of order 12, and this action is faithful. Thus, we have |Aut(Sh)| = 16 · 12 = 192. On the other hand, |Aut(Z4 )| = 2, therefore, |Aut(N (Z4 ))| = 42 · 2 · 3! = 192. Thus, we obtain that for the second main class Γ2 the groups of the 3-net and of the point graph coincide.

20

Aiso Heinze and Mikhail Klin

Fig. 3. The Shrikhande graph Sh

Finally, inspecting suitable subgroups of Aut(Γ1 ) and Aut(Γ2 ) we may prove that for n = 4 the results of the classiﬁcation of types, isotopy classes and loops do not provide surprises, that is, the number 2 appears in the corresponding row of Table 1 three more times. 4.5 The Case n = 5, Part a Counting the reduced squares in this case is still quite an easy job. It was Cayley, who found in [28] that Rn = 56. Here, proper loops appear for the ﬁrst time. There are exactly 15 strongly regular graphs with the parameters (v, k, λ, μ) = (25, 12, 5, 6). Only two of them are geometrical, in other words, there are exactly two main classes of Latin squares of order 5. One of these main classes, clearly, corresponds to the cyclic group of order 5. Denoting the corresponding Latin square graph as Γ1 = SRG(Z5 ), we get that |Aut(Γ1 )| = 52 · 4 · 6 = 600. A corresponding partial diﬀerence set over (Z5 )2 consists of a union of non-identity elements in three (arbitrary) subgroups of order 5 in (Z5 )2 . This is a rank 3 graph. There is, however, another classical way to get a rank 3 graph with the same parameters. Indeed, let us consider the Paley graph P (25). Its vertices are the elements of GF (25). Two vertices x, y are adjacent if and only if y − x is a square in GF (25). The graph P (25) is invariant with respect to the Frobenius group E25 Z12 , that is to the subgroup of index 2 in AGL(1, 25). This observation immediately implies that P (25) is a rank 3 graph. In fact, Aut(P (25)) has the order twice as large as E25 Z12 (add to it the automorphism of P (25) generated by the non-trivial automorphism of the ﬁeld GF (25)). Finally, elementary considerations show that Z12 acts transitively on non-identity elements from three (of six) proper cyclic subgroups of the additive group of GF (25). Thus, we conclude that Γ1 and P (25) may be regarded as the same Cayley graphs over E25 , that is, they are isomorphic.

Loops, Latin Squares and Strongly Regular Graphs

21

The main class corresponding to Γ1 produces just a single type, a single isotopy class and a single loop (which is a group). At this point we are faced with the necessity to consider another geometrical Γ2 . One can take any proper loop and obtain from it Γ2 . We, however, will prefer a more sophisticated procedure: to construct Γ2 in such a manner that will introduce its group Aut(Γ2 ). Note that Γ1 is a self-complementary graph, whereas Γ2 is not self-complementary. Indeed, otherwise, a pair (Γ2 , Γ 2 ) will provide another aﬃne plane of order 5 (diﬀerent from the classical one), contradicting the well-known fact that the projective plane of order 5 is unique (up to isomorphism). 4.6 The Case n = 5, Part b Let us ﬁrst consider the direct sum S4 + S3 and the direct product S4 × S3 of the symmetric groups S4 and S3 , acting as permutation groups of degree 7 and 12 respectively (see [59] for the discussion of these operations for permutation groups). Both groups are isomorphic as abstract groups and have order 144. For a short while it is more convenient for us to work with S4 + S3 . An easy exercise is to show that S4 + S3 has three diﬀerent subgroups of index 2, namely, A4 + S3 , S4 + A3 , and (S4 + S3 )pos . Here, for a permutation group (H, Ω), the group (H pos , Ω) denotes the intersection of H with the alternating group of Ω. In other words, H pos contains only even permutations from H. Consider now the following geometry presented in Fig. 4(a). Its set of lines contains three 4-element horizontal lines and four 3-element vertical lines. We will denote it by L3,4 (lattice of size 3 × 4) and we will call it brieﬂy a lattice. Its point graph Γ (L3,4 ) is depicted in Fig. 4(b). It is evident that Aut(L3,4 ) = Aut(Γ (L3,4 )) = S4 × S3 . Thus, we may use the more simple diagram in Part (a), even if we think of the lattice in terms of its point graph.

Fig. 4. The lattice L3,4 and its point graph

22

Aiso Heinze and Mikhail Klin

Now we consider the group G ∼ = (S4 + S3 )pos of order 72 as a subgroup of Aut(L3,4 ). We may use the following presentation for G: (0, 1, 2)(4, 5, 6)(8, 9, 10), (1, 2, 3)(5, 6, 7)(9, 10, 11), G= . (0, 4, 8)(1, 5, 9)(2, 6, 10)(3, 7, 11), (0, 5)(1, 4)(8, 9)(2, 6)(3, 7) Here the fourth generator corresponds to a simultaneous transposition of the ﬁrst two rows and the ﬁrst two columns of the lattice. Note that G in its action of degree 12 contains odd permutations. We now associate more geometrical objects to L3,4 and G. The graph Γ (L3,4 ) has anticliques of size 3, that is induced empty subgraphs of order 3. It is easy to see that altogether there are 24 such anticliques, which we will call triples. Let {0, 5, 10} be one of these triples. Then the stabilizer G{0,5,10} is a subgroup of order 6 in G: G{0,5,10} = (0, 5, 10)(1, 6, 8)(2, 4, 9)(3, 7, 11), (0, 5)(1, 4)(8, 9)(2, 6)(3, 7) . This implies that the set of triples is split into two orbits of length 12 under the action of G. Triples from the orbit {0, 5, 10}G will be called right triples, while ones from {2, 5, 8}G will be called left triples. For the reader’s convenience we list the right triples Tr and the left triples Tl below: Tr = {0, 5, 10}, {0, 6, 11}, {0, 7, 9}, {1, 4, 11}, {1, 6, 8}, {1, 7, 10}, {2, 4, 9}, {2, 5, 11}, {2, 7, 8}, {3, 4, 10}, {3, 5, 8}, {3, 6, 9} , Tl = {0, 5, 11}, {0, 6, 9}, {0, 7, 10}, {1, 4, 10}, {1, 6, 11}, {1, 7, 8}, {2, 4, 11}, {2, 5, 8}, {2, 7, 9}, {3, 4, 9}, {3, 5, 10}, {3, 6, 8} . It is easy to observe that the incidence structures with the point set [0, 11] and the line sets Tr , Tl are each partial linear spaces; that is, in each geometry any two points are joined by at most one line. Let us associate to these geometries a graph: vertices are the triples and two triples are adjacent if they are disjoint. It turns out that both graphs are isomorphic to Γ (L3,4 ). Moreover, we also get a structure of a lattice on the sets Tr and Tl . The diagrams of both lattices, which will be called right and left dual lattices respectively, are shown in Fig. 5. An analysis of both lattices shows that the group G may be described as the intersection of the automorphism groups of two incidence structures, namely G = Aut(([0, 11], Tr )) ∩ Aut(([0, 11], Tl )). This representation turns out to be a crucial observation in the coming description of our model for the graph Γ2 , which was implicitly introduced above. Recall that for n ≥ 5 the following concepts are equivalent: the main class of a Latin square Q of order n, its graph SRG(Q), the 3-net N (Q), and the

Loops, Latin Squares and Strongly Regular Graphs

23

Fig. 5. Left and right dual lattices

transversal design T D(Q). This means in particular, that each of these objects is uniquely reconstructible from the others. Here we prefer to describe ﬁrstly the incidence structure γ2 = T D(3, 5) which corresponds to Γ2 . First we have to decide which of the two dual lattices will play a “special role”. (Note that both selections provide diﬀerent, though isomorphic structures.) Thus, let us decide, e.g., the right one. Then the points of our structure are formed by the union P1 ∪ P2 of two sets, where P1 is the set of rows of the right dual lattice, while P2 is formed by 12 vertical pairs in the right dual lattice. Here, a vertical pair is a subset consisting of two triples in the same column. The points are divided into three disjoint groups, with each group containing a row from the right lattice and four vertical pairs which appear in the two remaining rows. Below is a list of three groups (as it was produced by a computer package COCO), which, in principle, may be obtained by simple routine enumeration. 0. {{0, 5, 10}, {1, 4, 11}, {2, 7, 8}, {3, 6, 9}}

8. {{0, 7, 9}, {1, 4, 11}}

1. {{0, 7, 9}, {1, 6, 8}, {2, 5, 11}, {3, 4, 10}}

9. {{0, 7, 9}, {3, 5, 8}}

2. {{0, 6, 11}, {1, 7, 10}, {2, 4, 9}, {3, 5, 8}}

10. {{0, 6, 11}, {3, 4, 10}}

3. {{0, 5, 10}, {1, 6, 8}}

11. {{2, 5, 11}, {3, 6, 9}}

4. {{1, 6, 8}, {2, 4, 9}}

12. {{1, 4, 11}, {3, 5, 8}}

5. {{0, 6, 11}, {2, 7, 8}}

13. {{1, 7, 10}, {3, 6, 9}}

6. {{0, 5, 10}, {2, 4, 9}}

14. {{1, 7, 10}, {2, 5, 11}}

7. {{2, 7, 8}, {3, 4, 10}} In what follows we will keep the numbers from [0,14] for the points from the set P = P1 ∪ P2 . Thus, in such a notation the groups are {0, 4, 9, 10, 14}, {1, 5, 6, 12, 13}, {2, 3, 7, 8, 11}. The set of lines is L1 ∪ L2 ∪ L3 . Here L1 includes just one line, which corresponds to the whole right lattice. L2 and L3 correspond respectively to

24

Aiso Heinze and Mikhail Klin

the triples in the right and left lattice. The incidence is described as follows: The unique line in L1 is incident to the three points in P1 . The line deﬁned by the right triple is incident to a row, in which this triple appears, and to two vertical pairs involving this triple. The line deﬁned by the left triple say {a, b, c} is incident to the three vertical pairs which are obtained as follows: • • •

ﬁnd in the right lattice three triples containing {a, b}, {a, c} and {b, c} respectively; each of these triples deﬁnes an “additional” vertical pair, that is two remaining triples in the same column of the right lattice; these additional pairs are incident to {a, b, c}.

For the reader’s convenience we show (in a new numeration of the elements of γ2 ) three lines; a line attributed to the whole lattice is {0, 1, 2}, a line for {0, 5, 10} is {0, 3, 6} and a line for {2, 5, 8} is {8, 10, 13}. We again provide a list of all lines obtained in such a manner, each line is represented as a 3-element set of incident points. 0. {0, 1, 2}

9. {1, 11, 14}

18. {3, 5, 14}

1. {0, 3, 6}

10. {2, 9, 12}

19. {6, 10, 11}

2. {1, 3, 4}

11. {0, 11, 13}

20. {3, 10, 12}

3. {2, 5, 10}

12. {2, 13, 14}

21. {3, 9, 13}

4. {2, 4, 6}

13. {8, 10, 13}

22. {4, 7, 13}

5. {0, 5, 7}

14. {7, 12, 14}

23. {4, 5, 8}

6. {0, 8, 12}

15. {6, 8, 14}

24. {6, 7, 9}

7. {1, 8, 9}

16. {5, 9, 11}

8. {1, 7, 10}

17. {4, 11, 12}

Now we have to check that the constructed incidence system γ2 is indeed a T D(3, 5). An inspection may be simpliﬁed, if we observe that our group G has three orbits on pairs of points from P, belonging to diﬀerent groups, namely: {0, 1}G of length 3, {0, 3}G of length 36 and {8, 10}G of length 36. Clearly, γ2 contains one line through each of the representatives. Counting the total number of lines, we multiply it by 3 and deduce that indeed γ2 is a partial linear space. Now it remains to describe Aut(γ2 ). By construction, G ≤ Aut(γ2 ). Denote G = Aut(γ2 ). Then sequentially using a well-known orbit-counting lemma, we may obtain the order of G. For this purpose we ﬁrst have to show that 0 and 3 belong to diﬀerent orbits of G. This seems to be a slightly cumbersome job. For example, we may get the Latin square graph Γ2 corresponding to γ2 , construct its subgraphs Γ2 (0) and Γ2 (1) induced by the neighbors of 0 and 1 respectively (recall that the vertices of Γ2 are lines of γ2 ). It turns that these subgraphs are non-isomorphic and may be distinguished by,

Loops, Latin Squares and Strongly Regular Graphs

25

Table 5. Parallel classes of lines of γ˜2 1. 2. 3. 4. 5.

{0, 1, 5, 6, 11} {2, 4, 17, 22, 23} {3, 8, 13, 19, 20} {7, 10, 16, 21, 24} {9, 12, 14, 15, 18}

1. 2. 3. 4. 5.

{0, 2, 7, 8, 9} {1, 4, 15, 19, 24} {3, 5, 16, 18, 23} {6, 10, 14, 17, 20} {11, 12, 13, 21, 22}

1. 2. 3. 4. 5.

{0, 3, 4, 10, 12} {1, 2, 18, 20, 21} {5, 8, 14, 22, 24} {6, 7, 13, 15, 23} {9, 11, 16, 17, 19}

e.g., a number of maximal 3-cliques, indeed Γ2 (0) has four maximal 3-cliques while Γ2 (1) has only one maximal 3-clique, cf. also [92]. Thus, we now get that the lines 0 and 1 are in diﬀerent orbits of G, and thus points 0 and 3 are also in diﬀerent orbits. Finally, we obtain (using each time suitable information about G): |G| = |G3 | · |3|G| | = 12 · |G3 | = 12 · |8G3 | · |G3,8 | = 36 · |G3,8 | = 36 · |0G3,8 | · |G0,3,8 | = 72 · |G0,3,8 |. Now an easy brute force inspection shows that the automorphism, which ﬁxes 0, 3 and 8, leaves all points of γ2 in place. Thus, |G| = 72 · 1 = 72, and therefore G = G. The introduced group G is indeed the full automorphism group of γ2 . We note that, like for Γ1 , there corresponds just one type and one isotopy class to the graph Γ2 . It turns out, however, that the graph Γ2 corresponds to the smallest case, where one main class contains a few non-isomorphic loops. Let us ﬁrst try to extract at least one loop from Γ2 . We consider the incidence structure γ˜2 , which is dual to γ2 . This is a 3-net with 25 points. It has three parallel classes of lines and each line contains ﬁve points. Let us (arbitrarily) distinguish the ﬁrst, second and third parallel class. We now have freedom to manipulate the order of the members of each class. Thus, ﬁnally, we label ﬁve blocks in each of the three classes by elements from [1, 5]. A suitable ordering is presented in Table 5. Now, the ﬁrst class deﬁnes the rows, the second the tables and the third the contents of the cell of the tables. We pick up a row i, a column j, ﬁnd the intersection {x} of the corresponding blocks, detect the unique block in the third class containing x and put its label on the intersection of row i and column j. Clearly, we obtain a Cayley table of a quasigroup. However, our lucky ordering provides us more: a loop. Its Cayley table is the following 1

2

3

4

5

2

1

4

5

3

3

5

1

2

4

4

3

5

1

2

5

4

2

3

1

L1

26

Aiso Heinze and Mikhail Klin

It turns out that using the same graph Γ2 we may also obtain the following Cayley tables of loops: 1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

2 1 5 3 4

2 1 4 5 3

2 1 4 5 3

2 3 4 5 1

3 4 2 5 1

3 4 5 1 2

3 4 5 2 1

3 5 2 1 4

4 5 1 2 3

4 5 2 3 1

4 5 1 3 2

4 1 5 3 2

5 3 4 1 2

5 3 1 2 4

5 3 2 1 4

5 4 1 2 3

L2

L3

L4

L5

Considering multisets of elements on a diagonal, we may claim that the loops L1 , L2 , and L5 each form a simple isomorphism class. It remains to distinguish combinatorially the loops L3 and L4 . For this purpose we will associate to each loop a certain set of permutations (see comments in Sect. 10). It is convenient to consider sets of permutations deﬁned by columns. Then we associate to L3 one permutation of type x51 and four permutations of type x2 x3 , while to L4 one of type x51 , one of type x2 x3 and three of type x5 . Thus, the loops L3 and L4 are also not isomorphic. 4.7 The Case n = 6 There are 12 main classes of squares of order 6, three of them have a transitive automorphism group. In particular, Aut(SRG(Z6 )) has order 62 · 2 · 6 = 432 and Aut(SRG(S3 )) has order 62 · 6 · 6 = 1296. The third main class with a transitive group will be presented in the next sections, its careful investigation is the central point in this article. Here we will consider one more main class, which has an intransitive automorphism group, though quite large and clear. Our consideration is inﬂuenced by [16], though we act in a diﬀerent fashion (see Sect. 10). For current purposes it will be convenient for us to use the language of association schemes. First we will present a scheme on a set Ω with four classes of valency 5, 5, 5, 20, which corresponds naturally to a 3-net. Three schemes with three classes, each obtained by merging of two classes of valency 5, will correspond to the diﬀerent types. Finally, a merging of three classes of valency 5 leads to the Latin square graph and its complement. Let us start with the group A5 of order 60 in its natural action on ﬁve points. The set of cardinality 36 comprises two orbits Ω1 and Ω2 deﬁned by the induced action of (A5 , [0, 4]). Let us consider a pentagon, that is a regular connected undirected graph of valency 2. Its automorphism group D5 consists of even permutations only. Therefore, an orbit of a pentagon under the action of A5 has cardinality 6. This is the set Ω1 , it is represented in Fig. 6. Let us consider an ordered pair of two disjoint 2-element subsets of [0, 4]. Clearly, all such pairs form a single orbit under the action of A5 . This is set Ω2 of cardinality 30.

Loops, Latin Squares and Strongly Regular Graphs

27

Fig. 6. The orbit Ω1 of A5

Now we need to deﬁne classes of valency 5 in the association scheme. Each such class corresponds to a graph of the form 6 ◦ K6 . Therefore, in order to represent such a speciﬁc binary relation, it is enough to deﬁne a partition of Ω into six sets of equal cardinality. The partition corresponding to R1 , contains the set Ω1 as one cell, ﬁve other cells are deﬁned by elements of [0, 4]. For example, a cell deﬁned by 4 contains ordered pairs ({0, 1}, {2, 3}), ({0, 2}, {1, 3}), ({0, 3}, {1, 2}), ({1, 2}, {0, 3}), ({1, 3}, {0, 2}), ({2, 3}, {0, 1}). The partition corresponding to R2 contains six cells, each is deﬁned by an element of Ω1 like P0 , ({0, 1}, {2, 4}), ({1, 2}, {0, 3}), ({2, 3}, {1, 4}), ({3, 4}, {0, 2}), ({0, 4}, {1, 3}) . Similarly, a partition corresponding to R3 is deﬁned by a typical cell P0 , ({2, 4}, {0, 1}), ({0, 3}, {1, 2}), ({1, 4}, {2, 3}), ({0, 2}, {3, 4}), ({1, 3}, {0, 4}) . It follows immediately from the deﬁnition that the partitions R1 , R2 , R3 are orthogonal, in a sense, that corresponding graphs are edge-disjoint.

28

Aiso Heinze and Mikhail Klin

3 Finally, R0 is a diagonal relation and R4 = Ω 2 \( i=0 Ri ). We get an amorphic association scheme M0 = (Ω, {R0 , R1 , R2 , R3 , R4 }) with four classes (cf. [45]), to which corresponds a certain main class of Latin squares of order 6. It follows from the construction that M0 is invariant with respect to the induced permutation group (A5 , Ω). It is easy to see, that in fact M0 is invariant with respect to the larger group of order 120, which is isomorphic to S5 . For this purpose let us consider A5 above as a subgroup of index 2 of S5 acting on the same set [0, 4]. Then S5 acts naturally on the set Ω2 . Let us consider the set Ω1 of all pairs consisting of a pentagon together with its complement. Clearly, |Ω1 | = |Ω1 |; moreover, because each pair of Ω1 contains exactly one pentagon from Ω1 , we get a natural bijection between the elements of Ω1 and Ω1 . The group S5 acts on Ω1 and therefore we may consider also the corresponding action of S5 on Ω1 . This action evidently preserves the relations R1 and R4 and the union R2 ∪R3 , while a more accurate glance detects that the relations R2 and R3 separately are also preserved by S5 . Let us now deﬁne one more permutation τ on Ω, which ﬁxes all elements of Ω1 , while each ordered pair of the form ({a, b}, {c, d}) is transposed with ({c, d}, {a, b}). Again very easy reasonings show that τ preserves the relations R1 and R4 and transposes the relations R2 and R3 . Moreover, τ commutes with all elements of S5 , therefore we may deﬁne the group S5 × τ which is isomorphic to S5 × S2 . Now we consider the association schemes M1 = (Ω; {R0 , R1 , R2 ∪R3 , R4 }), M3 = (Ω; {R0 , R1 ∪R2 , R3 , R4 }),

M2 = (Ω; {R0 , R1 ∪R3 , R2 , R4 }), M4 = (Ω; {R0 , R1 ∪R2 ∪R3 , R4 }).

It is again evident from the construction that Aut(M1 ) (and thus also Aut(M4 )) contains a subgroup isomorphic to S5 × S2 of order 240. Simple combinatorial arguments may be used to show that we already know all the automorphism groups, namely Aut(M1 ) = Aut(M4 ) = S5 × S2 and Aut(M0 ) = Aut(M2 ) = Aut(M3 ) = S5 . Moreover, each permutation from (S5 ×S2 )\S5 establishes an isomorphism between M2 and M3 , while M1 is not isomorphic to M2 and M3 . Thus, we have obtained a desired main class of Latin squares, which is split into two types, one corresponds to M1 and the other to M2 and M3 . One more attractive step would be a description of the isotopy classes in our main class, as well as the evident detection of at least one loop from this class. We, however, stop here, referring again to [16] for a nice description of a loop in terms of the icosahedron. (Note the well-known fact that the automorphism group of the icosahedron graph is exactly A5 × S2 .) Here we just present a multiplication table of a loop from the considered main class:

Loops, Latin Squares and Strongly Regular Graphs

1

2

3

4

5

6

2

1

6

3

4

5

3

6

1

5

2

4

4

3

5

1

6

2

5

4

2

6

1

3

6

5

4

2

3

1

29

5 The Remark of Barlotti and Strambach Now we are coming back to the main line of our presentation: the consideration of “group-like” quasigroups. Proposition 5. Consider the following Latin square Q6 (No 3.1.1 in [34]): 1

2

3

4

5

6

2

3

1

5

6

4

3

1

2

6

4

5

4

6

5

2

1

3

5

4

6

3

2

1

6

5

4

1

3

2

Then: (a) The main class of Q6 does not contain a group; (b) G = Aut(Γ (Q6 )) is a transitive permutation group of degree 36 and order 648 ; (c) G has a regular subgroup. Note that the quasigroup Q6 is a well-known object (see more details in Sect. 10). In particular, parts (a) and (b) of our claim can be extracted from many sources in the literature. Part (c) may be extracted from [96]. Nevertheless, it seems that, as an entity, the whole proposition appeared for the ﬁrst time in [50], where it was proved with the aid of a computer.

6 Computer Aided Answer The computer aided proof of part (c) in Proposition 5 was in a sense, a nonexpected by-product of a general project, which was undertaken by A. Heinze in his Ph.D. Thesis [50].4 4

Supervisors M. Klin and U. Knauer.

30

Aiso Heinze and Mikhail Klin

6.1 Computer-based Analysis of the Example with GAP The use of the computer system GAP (see [43]) allows us to obtain some set of generators for our group G of order 648. Recall that originally we started from a catalogue of strongly regular graphs [95], inspected a concrete graph with its automorphism group and checked whether this graph Γ is geometric, i.e., if Γ = SRG(Q). However, as soon as we agree that Q is our source of information we just create a description of SRG(Q), put it into GAP (more exactly into its share package GRAPE [94]) and then determine a desired set of generators for the group G. Moreover, using some of the simple routines, which are described in [50] (cf. also [53]), we easily observe that G has two (up to isomorphism) regular subgroups H1 and H2 (of order 36). One of these groups, say H2 , can be easily identiﬁed by GAP as H2 = S3 × S3 , that is the direct product of two symmetric groups of degree 3. For the whole group G it was not so easy to ﬁnd its structure. Thus, using ad hoc arguments we formulated the following question: Is it true that G ∼ = (S3 S3 )pos . In other words does G consist of all even permutations in the standard action of the wreath product of S3 with S3 on nine points? The immediate answer of GAP was “yes”. In a similar manner we guessed that for any point x ∈ Ω (here (G, Ω) is a transitive permutation group of degree 36) Gx ∼ = D9 , where Gx is the stabilizer of x in G and D9 is a dihedral group of degree 9 and order 18. Again, we obtained from GAP the conﬁrmation of our guess. At this moment we may complete to use GAP. We know the whole group G (as an abstract group), we know the subgroup D9 of G, which returns the stabilizer. Therefore, the action (G, Ω) is permutationally isomorphic to the action of G on the cosets modulo D9 . 6.2 Further Computer-based Analysis with COCO Now we are prepared to use another computer package COCO (COherent COnﬁgurations, cf. [39]). We would like to obtain more structural information about our permutation group (G, Ω). First we need to construct this action. For this purpose we have to create for COCO a description of G and a description of its subgroup D9 as an input. We describe G as a group of degree 9 acting on the set [0,8] with the aid of generators. Thus G = g1 , g2 , g3 , g4 , where g1 = (0, 1, 2, 3, 4, 5, 6, 7, 8), g2 = (0, 3, 6), g3 = (0, 3)(1, 4), g4 = (0, 1)(3, 4, 6, 7). We present extra comments on this set below. Instead of a description of the subgroup D9 , COCO requires (and this is in fact one of its main paradigms) a relational structure S on the same set [0,8], preferably as simple as possible, such that Aut(S) ∩ G = D9 . It is clear how to

Loops, Latin Squares and Strongly Regular Graphs

31

Fig. 7. Cycle C9 of length 9

manage a fulﬁllment of such a requirement. Consider an unordered cycle C9 of length 9 and let us set S := C9 (we mean here by a cycle a regular connected graph of valency 2). It is convenient to select a canonical labeling of vertices in our copy of C9 like in Fig. 7. All automorphisms from D9 = Aut(C9 ) are even permutations. Thus, we can select in advance (knowing our copy of C9 ) such a representation of G that our general requirement above, that is Aut(C9 ) ∩ G = D9 , will be substituted by a simpler condition: D9 < G. This provides the way to obtain a description of G. Consider the disconnected graph 3 ◦ K3 as shown in Fig. 8. Evidently D9 < Aut(3 ◦ K3 ) and G ∼ = (Aut(3 ◦ K3 ))pos = (S3 S3 )pos . Finally, we just need to check that g1 , g2 , g3 , g4 ∈ (S3 S3 )pos and moreover that (S3 S3 )pos = g1 , g2 , g3 , g4 . Thus, we provide the promised justiﬁcation of our generators. Now we are ready to discuss the output of COCO. It consists of the following pieces of information: • • • •

the generators g1 , g2 , g3 , g4 of the induced action of G on the set Ω of all diﬀerent copies of C9 with respect to permutations from G; this action has degree 648 18 = 36; the 2-orbits of this action and the corresponding subdegrees: 1, 1, 1, 6, 9, 9, 9; the intersection numbers of a corresponding association scheme M = (Ω, 2-orb(G, Ω)) with six classes; all mergings of classes of this association scheme.

In particular, we obtain that there exist three non-Schurian mergings with two classes with valencies 1, 15, 20. It turns out that these three mergings correspond to three (isomorphic) copies of the strongly regular graph

Fig. 8. Graph 3 ◦ K3

32

Aiso Heinze and Mikhail Klin

Fig. 9. The auxiliary graph Δ = 3 ◦ K3

Γ = SRG(Q6 ) with parameters v = 36, k = 15, λ = 6, μ = 6. (One may again use GAP to conﬁrm this information.) Thus, we have taken a ﬁrst step towards understanding our graph Γ . More exactly we approach a computer free explanation of some results obtained by the conjunction of GAP and COCO. Consider the auxiliary graph Δ = 3 ◦ K 3 as in Fig. 9. Here the sign ⇐⇒ in Fig. 9 means a set of nine edges of a complete bipartite graph K3,3 . An easy combinatorial exercise shows that Δ has exactly 3 · 3 · 2 · 2 · 2 = 72 spanning subgraphs which are Hamiltonian cycles in Δ. Our initial canonical copy of C9 is shown once more in Fig. 10. There are two orbits of length 36 in the action of G on these 72 cycles. Recall that G is a “half” of Aut(Δ), while all permutations in D9 are even. Thus, we have the typical Buridanian Donkey problem: How do we select one of these two isomorphic orbits? The criterion which we are using is to take the orbit which includes our canonical cycle C9 . Now, analyzing the output of COCO, we are obtaining representatives of neighbors of a canonical C9 in all six 2-orbits of (G, Ω) (see Fig. 11). The huge numbers in Fig. 11 show the corresponding subdegrees of the 2-orbits. Now we have three possibilities to merge a relation of valency 6 with

Fig. 10. Copy of C9 as Hamiltonian cycle in Δ

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Fig. 11. Representatives of neighbors of a canonical C9 in all six 2-orbits of (G, Ω)

a relation of valency 9. Each time we reach a desired strongly regular graph of valency 15. We call the obtained description an explanation of the graph Γ . We mean that we, in principle, may now explain post factum vertices and edges of Γ without the use of a computer. In the next section we present a real computer free interpretation: such way of reasonings where everything, including proofs, are independent of routine computations.

7 Computer Free Interpretation This section is the central one in our presentation. We are trying to make it self-contained as much as it is possible. 7.1 General Idea Starting from a given Latin square Q (of order 6) we are getting a corresponding 3-net N (Q) with 36 points and 18 lines. This is a partial geometry. Recall that the dual incidence structure is also a partial geometry namely a transversal design T D(3, 6). It turns out that in our case of Q6 it is much easier to work with T D(3, 6) than with N (Q6 ). The reason is that we have a very transparent explicit description of the points of T D(3, 6) and still above an implicit description

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of lines (= points of the net). This description will be enough to check that all axioms of T D(3, 6) are satisﬁed and that we may justify all requested properties. 7.2 Axioms of T D(3, 6) We need an incidence structure S = (P, L) such that: (i) |P| = 18, |L| = 36. (ii) P = P1 ∪ P2 ∪ P3 is a partition of P into three classes of size 6, which are called groups of S. (iii) Each block has cardinality 3. (iv) Every unordered pair of distinct points either is contained in exactly one group, or is contained in exactly one block, but not both. 7.3 Description of a Model Our starting basic object will be again the graph Δ = 3 ◦ K3 , as it was depicted in Fig. 9. By a partial 1-factor of Δ we mean a set of three edges of Δ which form a 1-factor of a graph K3,3 (a subgraph of Δ which is depicted by the sign “⇐⇒” in Fig. 9). An example of such a 1-factor is provided in Fig. 12. Clearly we have 3 · 3! = 18 partial 1-factors. The set P of our incidence structure S consists of all such objects. The blocks are exactly 36 selected copies of C9 from our orbit Ω as they were deﬁned in Sect. 6.2. It is clear from our construction that each Hamiltonian cycle can be split into three partial 1-factors. This provides us with a natural incidence relation between points and blocks. 7.4 Fulﬁllment of Axioms It is immediately clear from our deﬁnition that the Axioms (i) and (iii) are satisﬁed. The edge set of Δ is naturally split into edge sets of three copies of K3,3 (i.e., three signs ⇐⇒ in Fig. 9). Each copy of K3,3 evidently contains six partial 1-factors. Thus, we have a partition of P which is requested in (ii). It remains to check if Axiom (iv) is satisﬁed. More exactly, we have to take two

Fig. 12. Example of a 1-factor of Δ

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Fig. 13. Two partial 1-factors of Δ

Fig. 14. Hamiltonian cycles C , C of Δ

partial 1-factors in distinct groups and to show that they appear in exactly one cycle in Ω. Note that the group G acts transitively on such pairs. This implies that we can present a purely pictorial proof. Two partial 1-factors are depicted in Fig. 13. It is easy to see that together they appear in exactly two Hamiltonian cycles C , C of Δ, see Fig. 14. Let us now ﬁnd an arbitrary permutation g which sends C to C , for example g = (4, 7)(5, 8)(3, 6). The set of all such permutations is a coset D9 · g in S9 , which evidently consists of odd permutations only. Therefore, exactly one of C and C belongs to our orbit Ω. We are done (and we do not care to know, which of the two is actually in Ω). Therefore, indeed, S is a model of a transversal design T D(3, 6). 7.5 The Group Aut(S) It is clear from the construction that G = Aut(S) ≥ G. We need to prove that, in fact, |Aut(S)| = |G|. This evidently will imply that Aut(S) = G. For this part of the proof we prefer to use a list of points and lines of S. We obtained it using COCO, though any alternative hand or computer tools may be exploited by the reader. Below we provide a list of elements of P labeled from 0 to 17, each label corresponds to a partial 1-factor of Δ (see Table 6). Note that we have the following groups (partition of P): {0, 2, 6, 10, 15, 16}, {1, 3, 5, 7, 9, 14}, {4, 8, 11, 12, 13, 17}.

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Aiso Heinze and Mikhail Klin Table 6. Elements of P and L

Elements of P 0. {{1, 2}, {4, 5}, {7, 8}} 1. {{0, 8}, {2, 3}, {5, 6}} 2. {{1, 5}, {2, 4}, {7, 8}} 3. {{0, 2}, {3, 8}, {5, 6}} 4. {{0, 1}, {3, 4}, {6, 7}} 5. {{0, 5}, {2, 6}, {3, 8}} 6. {{1, 8}, {2, 4}, {5, 7}} 7. {{0, 8}, {2, 6}, {3, 5}} 8. {{0, 4}, {1, 3}, {6, 7}} 9. {{0, 5}, {2, 3}, {6, 8}} 10. {{1, 2}, {4, 8}, {5, 7}} 11. {{0, 7}, {1, 3}, {4, 6}} 12. {{0, 1}, {3, 7}, {4, 6}} 13. {{0, 4}, {1, 6}, {3, 7}} 14. {{0, 2}, {3, 5}, {6, 8}} 15. {{1, 5}, {2, 7}, {4, 8}} 16. {{1, 8}, {2, 7}, {4, 5}} 17. {{0, 7}, {1, 6}, {3, 4}}

0. {0, 1, 4} 1. {0, 5, 11} 2. {2, 3, 4} 3. {3, 6, 12} 4. {1, 6, 13} 5. {0, 13, 14} 6. {2, 7, 13} 7. {3, 8, 15} 8. {0, 7, 8} 9. {2, 9, 11} 10. {5, 10, 12} 11. {8, 10, 14} 12. {6, 8, 9} 13. {1, 10, 17} 14. {4, 7, 10} 15. {4, 14, 15} 16. {4, 5, 6} 17. {3, 10, 11}

Elements of L 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

{6, 7, 17} {1, 11, 15} {0, 3, 17} {7, 12, 15} {5, 16, 17} {2, 5, 8} {9, 15, 17} {1, 8, 16} {9, 10, 13} {1, 2, 12} {3, 13, 16} {2, 14, 17} {4, 9, 16} {6, 11, 14} {0, 9, 12} {12, 14, 16} {7, 11, 16} {5, 13, 15}

In Table 6 we also provide a list of elements of L where each line in L is identiﬁed with a 3-element subset of P as it is labeled from the left. Now we would like to ﬁnd a pointwise stabilizer H = G0,1 of the points 0, 1 in Aut(S). Clearly, H leaves also point 4 in place. In what follows we will associate the gray color to the ﬁrst group, the black color to the second and the white color to the third group. Besides line {0, 1, 4} there are exactly 15 lines each of which contains exactly one of the elements 0, 1, 4. Depending on the element, we will also attribute a color to such a line. Thus, we come to an auxiliary structure with colored vertices and edges as follows (see Fig. 15, white lines are thin). The automorphism group H ∗ of this structure is generated by two permutations: h1 = (3, 7, 9)(2, 10, 16)(8, 12, 17) and

h2 = (5, 14)(6, 15)(11, 13),

Fig. 15. Auxiliary structure with colored vertices and edges

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that is H ∗ has order 6. However, for {3, 6, 12} ∈ L we have {3, 6, 12}h2 = / H and |H| ≤ 3. The reader can easily check {3, 15, 12} ∈ / L. Therefore h2 ∈ that, in fact, h1 ∈ G (and thus in H), it corresponds to an automorphism (1, 4, 7)(2, 5, 8)(0, 3, 6) of Δ. Therefore, |H| = 3. Now we obtain |G| = |G0 |·|0G | = 18·|G0 | = 18·|G0,1 |·|1G0 | = 18·12·|G0,1 | = 18·12·3 = 648. Therefore, G = Aut(S) = G. 7.6 Transversal Designs of Groups of Order 6 Recall now that according to Theorem 1 |Aut(SRG(Z6 ))| = 62 · 2 · 6 = 432 and

|Aut(SRG(S3 ))| = 62 · 6 · 6 = 1296.

Evidently, there are exactly two distinct groups of order 6. Therefore, because the order of G is not equal to 432 or 1296, we conclude with the aid of Corollary 3, that our transversal design S is not coming from a group. In other words, any loop corresponding to this design is indeed a proper loop. 7.7 Regular Subgroup for the Loop Case Now we have to ﬁnd a regular subgroup in the action (G, P). For this purpose we may use the action of G on the graph Δ. It is suﬃcient to ﬁnd a subgroup H of G such that all h ∈ H, h = e do not preserve any of the 36 copies of the cycle C9 , which form the set L. First we will present one copy of such a subgroup. Let K1 := (0, 3, 6), (0, 3)(2, 5), K2 := (1, 4, 7), (1, 4)(2, 5) and deﬁne H2 := K1 × K2 . Then: (a) (b) (c) (d)

∼ S3 , moreover H2 ∼ K1 ∼ = K2 = = S3 × S3 ; |H2 | = 36; H2 ≤ G; there is no copy of C9 in L which is preserved by H2 .

This means that we proved (more or less without essential use of a computer) that we have an example which provides a positive answer on the question of Barlotti-Strambach. In fact, our group G contains one more copy of a regular subgroup (up to isomorphism). Let now a = (0, 3, 6), b = (1, 4, 7), c = (0, 1, 3, 4)(6, 7) and deﬁne H1 := a, b, c. It is easy to check that c−1 ac = b and c−1 bc = a2 . Therefore, a, b ∼ = (Z3 )2 is a normal subgroup of H1 . ∼ Thus, H1 = (Z3 )2 Z4 is a group of order 36. This group is non-isomorphic to H2 , because H2 does not have elements of order 4. Clearly H1 ≤ G. Each element of H1 ﬁxes the vertices 2, 5, 8 of Δ. Taking into account that each non-identical permutation in D9 has at most one ﬁxed

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point we have shown that H1 acts regularly on the set L. Thus, G indeed has at least two regular subgroups. In fact, H1 , H2 are the only regular subgroups in G (up to isomorphism). This fact was established, using GAP (see details in [50]).

8 Exceptional Quasigroup Re-visited Recall our methodology. We started from the exceptional quasigroup Q6 , constructed its graph Γ , found the group G = Aut(Γ ) and after that restarted our consideration again. Thus we have assumed to pretense of not remembering Q6 . All we know is the group G, the action of G on future points P and lines L and the incidence in S. We check that indeed S is a T D(3, 6) and G = Aut(S). Now we would like to come back to Q6 . Clearly, there are many options to establish a quasigroup belonging to S; but we will reconstruct Q6 exactly. In principle, the methodology of our behavior is clear. As soon as we have S, ﬁrst we should attribute to it three groups that will be the rows, columns and elements of a future Latin square (we have here 3! options). After that we label the elements of each group by the numbers from the set {1, 2, 3, 4, 5, 6} – altogether (6!)3 options. Finally, we read our list of lines in L and ﬁll, step by step each of the 36 cells of a square. We, of course, may use for this purpose labels of elements from P and L. It turns out that it is a bit tricky to ﬁnd a possible labeling of the elements from P and L. However, by a simple trial-and-error method we found a labeling which is suitable. In the following Table 7 we did not label the rows, columns and elements of the Latin square by numbers from the set {1, 2, 3, 4, 5, 6} but by permutations of the symmetric group S3 . As we will see later, there is a correspondence between our quasigroup and the group S3 . Table 7 gives a one-to-one correspondence between elements of each of the three groups forming P (in the labeling of Table 6) and the permutations of S3 . To create the quasigroup Q6 we have to deﬁne the products of the rows and the columns. For example, take the second row – the permutation (a, b, c) – and the fourth column – the permutation (a, b). The corresponding numbers Table 7. Labeling of points in P Rows Permutation e (a, b, c) (a, c, b) (a, b) (b, c) (a, c)

No. in P 4 13 11 8 12 17

Columns Permutation e (a, b, c) (a, c, b) (a, b) (b, c) (a, c)

No. in P 15 6 0 2 10 16

Symbols Permutation e (a, b, c) (a, c, b) (a, b) (b, c) (a, c)

No. in P 14 5 1 3 7 9

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Table 8. Created Cayley table of the exceptional quasigroup Q6 e (a, b, c) (a, c, b) (a, b) (b, c) (a, c)

e e (a, b, c) (a, c, b) (a, b) (b, c) (a, c)

(a, b, c) (a, b, c) (a, c, b) e (a, c) (a, b) (b, c)

(a, c, b) (a, c, b) e (a, b, c) (b, c) (a, c) (a, b)

(a, b) (a, b) (b, c) (a, c) (a, b, c) (a, c, b) e

(b, c) (b, c) (a, c) (a, b) e (a, b, c) (a, c, b)

(a, c) (a, c) (a, b) (b, c) (a, c, b) e (a, b, c)

of points in P are 13 and 2. Now, to deﬁne the product of the multiplication (a, b, c) · (a, b) we have to ﬁnd the unique line in L that contains the points 13 and 2. Table 6 shows that this is the line No. 6 with points {2, 7, 13}. Hence the product of (a, b, c) · (a, b) is point No. 7 which corresponds to the element (b, c) as Table 7 shows. Repeating this procedure for all combinations of rows and columns we obtain the Cayley table of our exceptional quasigroup Q6 as it is given in Table 8. Note that in the framework of our entire presentation it is natural to use elements in S3 as labels (in fact S3 ∼ = D3 , cf. the notation in the next section). The reader should check that up to the used notation we obtain exactly our starting Cayley table for Q6 . So, in a sense, our Q6 is interpreted as a twisted dihedral group D3 of order 6. It remains to describe the form of this twist. Denote by x ◦ y the multiplication in Q6 , while xy is the usual multiplication in D3 . Let us select a “standard” permutation s = (a, b, c), a generator of a cyclic group of order 3. The permutation s will play the role of a distinguished permutation from D3 . Then we obtain ⎧ xy, if sign(x) = sign(y) = 1, ⎪ ⎪ ⎨ x ◦ y = xy, if sign(x) · sign(y) = −1, ⎪ ⎪ ⎩ xys, if sign(x) = sign(y) = −1. Recall that sign(x) = 1, if x is even and sign(x) = −1 otherwise. We have now a simple explanation of Q6 .

9 More Examples In this section we will outline an inﬁnite series of examples of proper quasigroups, each of which provides a positive answer on the question of BarlottiStrambach. Such a quasigroup Q2p will be described for each prime p, p ≡ 3 (mod 4). Our previous example Q6 is the ﬁrst member of the series corresponding to p = 3. The methodology of our presentation follows exactly the line which was developed for Q6 . We will not try to reach the complete level of rigor, omitting part of details in a few of the steps if they require some routine computations or more sophisticated reasonings. We will also illustrate

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each step by a full-length consideration of the second member of the series corresponding to p = 7. 9.1 Some Preliminary Observations Throughout this section p as a rule denotes a prime such that p ≡ 3 (mod 4). Let Z3p be an additive cyclic group of order 3p which is deﬁned by the addition modulo 3p. By (Z3p , Z3p ) we mean a regular representation of Z3p which is generated by a permutation g1 : x → x+1 (mod 3p). We denote by C3p a cycle of length 3p, that is a connected graph of valency 2 with 3p vertices. As a rule, we mean a canonical copy of C3p , that is a Cayley graph Cay(Z3p , {1, 3p−1}). It is clear that Aut(C3p ) = D3p , where D3p is a dihedral group of order 6p and degree 3p. We also have D3p = g1 , t, where t : x → −x (mod 3p). It turns out that Z3p and D3p have an imprimitivity system consisting of three blocks of size p, namely X0 , X1 , X2 , where Xi = {x ∈ Z3p |x ≡ i (mod 3)} for i = 0, 1, 2. For example, for p = 7 we get the blocks X0 = {0, 3, 6, 9, 12, 15, 18}, X1 = {1, 4, 7, 10, 13, 16, 19}, X2 = {2, 5, 8, 11, 14, 17, 20}. Let Σ = Cay(Z3p , {3, 3p − 3}). It is easy to see that Σ is a disconnected regular graph of valency 2 consisting of three connected components, each of which is isomorphic to a cycle Cp . For example, in the case of p = 7, Σ is depicted in Fig. 16. Clearly, Aut(Σ) ∼ = S3 Dp is a wreath product of the groups S3 and Dp , a group of order 6 · (2p)3 = 48p3 . Let Δ = Cay(Z3p , X1 ∪ X2 ) be a complete regular 3-partite graph of valency 2p. The graph Δ is isomorphic to 3 ◦ Kp , i.e. the disjoint union of three complete graphs with p vertices. Note that for p > 3 the group Aut(Δ) = Aut(Δ) ∼ = S3 Sp , and thus is strictly larger than S3 Dp . It is important to note that D3p is a subgroup of S3 Dp . Note also that the cycles of length 2. Thus, t is an permutation t is an involution which has 3p−1 2 even permutation, as well as g1 . Therefore, D3p consists of even permutations only. On the other hand, the group Dp and therefore S3 Dp contain odd permutations, for example, each involution in Dp has p−1 2 cycles of length 2. Therefore, G = (S3 Dp )pos , a subgroup of all even permutations in Aut(Σ),

Fig. 16. Graph Σ for p = 7

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has index 2. This group G of order 24p3 is one of the main heroes of our presentation in this section. 9.2 Deﬁning Points and Lines Now we have to make some combinatorial preparations in order to deﬁne our future incidence system S = Sp which will turn out to be T D(3, 2p). Let us consider a subgraph F of the canonical graph C3p which consists of all edges in C3p that join a vertex from X0 to a vertex of X1 . For p = 7 such a subgraph is depicted in Fig. 17. Note that Aut(F ) ∩ D3p is a subgroup isomorphic to Dp ≤ D3p which has index 3 in D3p . Here Dp = g13 , tg1 . We must now determine Aut(F ) ∩ G. If, for example, p = 7 we get Aut(F )∩G = g13 , tg1 , z2 , i2 , where z2 = (2, 5, 8, 11, 14, 17, 20), i2 = (3, 18)(6, 15)(9, 12)(4, 19)(7, 16)(10, 13). (Index 2 in z2 and i2 refers to X2 , which plays a special role in the deﬁnition of these permutations.) In general, deﬁning similar permutations, we get that Aut(F ) ∩ G is a group of order 4p2 . At this stage we deﬁne P = F G as the orbit of the action of G on all images of F under g ∈ G, that is P = {F g |g ∈ G}. Because the action (G, P) is transitive we see that 24p3 |G| = |P| = = 6p. |Aut(F ) ∩ G| 4p2 It is clear that the union of all 1-factors from P coincides with the edge set of the graph Δ. This set consists of 12 · 3p · 2p = 3p2 edges. Each 1-factor has exactly p edges. Thus, we ﬁnally prove that each edge of Δ appears in exactly two 1-factors from P. Now we deﬁne the set L of lines. Again we consider an image of a certain g G = {C3p |g ∈ G}. Taking structure under the action of G. This gives us L = C3p into account that Aut(C3p ) = D3p ≤ G, we conclude that |L| =

24p3 |G| = = 4p2 . |D3p | 6p

Fig. 17. Subgraph F of C21

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9.3 Constructing a Transversal Design The deﬁnition of an incidence system S = Sp is now evident. P is the set of points, L is the set of lines, the incidence coincides with the set-theoretical inclusion of 1-factors to a cycle. It is clear that P is naturally partitioned into three sets, namely P01 , P12 and P02 where Pij is the set of all 1-factors in P between vertex sets Xi and Xj . Proposition 6. The incidence structure Sp is a transversal design T D(3, 2p). Proof. We have |P| = 6p, |L| = 4p2 and it is clear that each element of L can be regarded as a 3-element subset of P. By deﬁnition, there is no line which goes through two distinct points in the same group. Due to vertex-transitivity 2 of G, each point is incident to the same number of lines, namely to 4p6p·3 = 2p lines. It remains then to check that through any two points from distinct groups there is exactly one line which goes through these groups. Note that the group G acts transitively on the ordered pairs of 1-factors from distinct groups. To prove this, consider any concrete 1-factor F , the group Aut(F ) ∩ G of order 4p2 , an arbitrary factor F in another group and show that the group Aut(F ) ∩ Aut(F ) ∩ G has order p and, thus, index 4p in Aut(F ) ∩ G. For example, in the case p = 7 we obtain Aut(F ) ∩ Aut(F ) ∩ G = g13 , where F and F are depicted in Fig. 18 (F contains {0, 1}, while F contains {1, 2}). Hence, for the proof it is enough to consider a canonical pair of 1-factors. Let F be the 1-factor of C3p which contains the edge {0, 1} while F is a 1-factor of C3p that contains edge {1, 2}. (These 1-factors are depicted in

Fig. 18. 1-factors F and F of Δ

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Fig. 18.) Clearly, the canonical cycle C3p goes through F and F . Our goal is to prove that C3p is the only one in L. Let H = {g ∈ G|(F g ∈ {F, F }) ∧ ((F )g ∈ {F, F })} be the stabilizer of the subset {F, F } in G. It follows from the above considerations that 3 |G| |H| = 1 6p·4p = 24p 12p2 = 2p. 2

Now consider the subgroup g13 , g1−1 tg1 . Denote t1 = g1−1 tg1 and check that t1 : x → 2 − x (mod 3p). From this it follows that {0, 1}t1 = {1, 2}, {1, 2}t1 = {0, 1} and more generally, that F and F are permuted by t1 . Thus g13 , t1 is a subgroup of H. Taking into account that |g13 , t1 | = 2p, we see that H = g13 , t1 . Suppose now that C ∈ L is another cycle from L which goes through F and F . Then, according to the deﬁnition of L, there exists a permutation g ∈ G such that (C3p )g = C . Note that C3p = F ∪ F ∪ F , while C = F ∪ F ∪ F for suitable F (here F is the remaining 1-factor of the canonical C3p which includes the edge {2, 3}). Without loss of generality we may assume, that g preserves the subset F ∪ F . Therefore, g belongs to H. Now it easy to check that H ≤ D3p = Aut(C3p ). Thus, if g preserves C3p and (F ∪ F )g = F ∪ F we have that (F )g = F . Therefore, F = F and C = C3p . (We invite the reader to consider the case p = 7 with the permutation t1 = (0, 2)(3, 20)(4, 19)(5, 18)(6, 17)(7, 16)(8, 15)(9, 14)(10, 13)(11, 12).) This shows that the incidence structure S is indeed a transversal design T D(3, 2p). 9.4 Automorphisms of the Design We are now describing a simple combinatorial procedure, which is a generalization of the process of construction of the auxiliary structure presented in Sect. 7.5 in Fig. 15. It can be applied for an arbitrary transversal design dual to a 3-net. Assume {A, B, C} is the partition into the groups of the point set of an arbitrary transversal design T D(3, 2p). Let Pα ∈ A, Pβ ∈ B, Pγ ∈ C. First we assign to a 3-element subset {Pα , Pβ , Pγ } a monochromatic undirected graph M = M ({Pα , Pβ , Pγ }) with the vertex set P as follows: {Px , Py } is an edge in M if and only if {Px , Py , Pz } is a line in T D(3, 2p) for z ∈ {α, β, γ}. We can also color the vertices and edges of M by the colors from {A, B, C}. To a vertex of P we assign the color corresponding to the group containing this vertex. To an edge {Px , Py } we assign the color of Pz , where {Px , Py , Pz } is as above. The resulting coloring of the auxiliary graph M will be denoted by C = C(Pα , Pβ , Pγ ). The next lemma follows easily from our deﬁnitions. Lemma 4. (a) The graph M ({Pα , Pβ , Pγ }) is a regular graph of valency 2, therefore each connectivity component is a cycle. (b) The graph M ({Pα , Pβ , Pγ }) is invariant with respect to the setwise stabilizer of the subset {Pα , Pβ , Pγ } in the group Aut(T D(3, 2p)).

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(c) The colored graph C(Pα , Pβ , Pγ ) is invariant with respect to the pointwise stabilizer of the points Pα , Pβ , Pγ in the group G. Remark. 1. Of course, in the formulation of Lemma 4 the number p is an arbitrary integer. 2. The graphs M and C may be deﬁned and exploited in a similar manner for an arbitrary partial linear space with lines of size 3. In particular, in the case of Steiner triple systems, the graphs M form a set of important invariants, see Sect. 4.3.2 of [30]. Our goal is now to describe the full automorphism group G = Aut(S), where S is the transversal design deﬁned in Sect. 9.2. Recall that the group G, and therefore also G acts transitively on the vertices and lines of S. Moreover, by deﬁnition the stabilizer of an arbitrary line from S in the group G is the dihedral group D3p of order 6p. In its action on points the group D3p has one orbit of length 3p and p orbits of length 3. (To observe this fact, note that each edge of a canonical copy of C3p is contained in exactly one of the above deﬁned factors F, F , F , as well as in one more factor. These 3p factors form the orbit of length 3p.) One of the orbits of length 3 exactly consists of the three points forming the considered line. The pointwise stabilizer of these three points in G is the unique cyclic group of order p which is contained in D3p . This cyclic group has in its action on P three orbits of length p and 3p ﬁxed points, including the points F, F , F . More advanced reasonings show that the auxiliary graph M (l) for an arbitrary line l from S, and in particular for l = {F, F , F } has one cycle of length 3p (the one induced by the above orbit of length 3p) and p−1 2 cycles of length 6 (each cycle corresponds to two of the above mentioned orbits of length 3). To avoid further sophistication we will now restrict ourselves to the consideration of the case p = 7. Using COCO, for a certain set of generators of the group G we create the following list of 1-factors from P (see Table 9). For each of the following pictures we will use the labeling of the elements from P created by COCO. In particular, the labels 0, 1, 2 correspond to our 1-factors F, F , F respectively. First we depict the auxiliary graph C(F, F , F ) (see Fig. 19 on page 47, conventions about the colors are the same as in Fig. 15). It is easy to see that the automorphism group Aut(C(F, F , F )) is isomorphic to the direct product of the group of order 7 acting semiregularly on the vertex set of the cycle of length 21 with the group of order 48 acting on the vertices of the three hexagons. The latter group is intransitive with three orbits of length 6 corresponding to the subsets (groups) A, B, C of the vertex set P. We now take vertex 12 from group C = P02 and describe the pointwise stabilizer G0,1,12 . For this purpose we will depict one more auxiliary graph C(F, F , F ), where F is the 1-factor with label 12 in our list, see Table 9.

Loops, Latin Squares and Strongly Regular Graphs Table 9. Points and lines of S in the case p = 7 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41.

Points of S {{0, 1}, {3, 4}, {6, 7}, {9, 10}, {12, 13}, {15, 16}, {18, 19}} {{1, 2}, {4, 5}, {7, 8}, {10, 11}, {13, 14}, {16, 17}, {19, 20}} {{0, 20}, {2, 3}, {5, 6}, {8, 9}, {11, 12}, {14, 15}, {17, 18}} {{0, 19}, {1, 3}, {4, 6}, {7, 9}, {10, 12}, {13, 15}, {16, 18}} {{0, 1}, {3, 19}, {4, 18}, {6, 16}, {7, 15}, {9, 13}, {10, 12}} {{1, 17}, {2, 16}, {4, 14}, {5, 13}, {7, 11}, {8, 10}, {19, 20}} {{0, 17}, {2, 6}, {3, 20}, {5, 9}, {8, 12}, {11, 15}, {14, 18}} {{1, 20}, {2, 4}, {5, 7}, {8, 10}, {11, 13}, {14, 16}, {17, 19}} {{0, 2}, {3, 5}, {6, 8}, {9, 11}, {12, 14}, {15, 17}, {18, 20}} {{0, 16}, {1, 6}, {3, 19}, {4, 9}, {7, 12}, {10, 15}, {13, 18}} {{0, 19}, {1, 18}, {3, 16}, {4, 15}, {6, 13}, {7, 12}, {9, 10}} {{1, 2}, {4, 20}, {5, 19}, {7, 17}, {8, 16}, {10, 14}, {11, 13}} {{0, 20}, {2, 18}, {3, 17}, {5, 15}, {6, 14}, {8, 12}, {9, 11}} {{0, 4}, {1, 3}, {6, 19}, {7, 18}, {9, 16}, {10, 15}, {12, 13}} {{0, 4}, {1, 18}, {3, 7}, {6, 10}, {9, 13}, {12, 16}, {15, 19}} {{0, 14}, {2, 9}, {3, 17}, {5, 12}, {6, 20}, {8, 15}, {11, 18}} {{1, 20}, {2, 19}, {4, 17}, {5, 16}, {7, 14}, {8, 13}, {10, 11}} {{1, 17}, {2, 7}, {4, 20}, {5, 10}, {8, 13}, {11, 16}, {14, 19}} {{0, 5}, {2, 18}, {3, 8}, {6, 11}, {9, 14}, {12, 17}, {15, 20}} {{0, 13}, {1, 9}, {3, 16}, {4, 12}, {6, 19}, {7, 15}, {10, 18}} {{0, 16}, {1, 15}, {3, 13}, {4, 12}, {6, 10}, {7, 9}, {18, 19}} {{0, 2}, {3, 20}, {5, 18}, {6, 17}, {8, 15}, {9, 14}, {11, 12}} {{0, 5}, {2, 3}, {6, 20}, {8, 18}, {9, 17}, {11, 15}, {12, 14}} {{1, 5}, {2, 4}, {7, 20}, {8, 19}, {10, 17}, {11, 16}, {13, 14}} {{0, 17}, {2, 15}, {3, 14}, {5, 12}, {6, 11}, {8, 9}, {18, 20}} {{0, 7}, {1, 6}, {3, 4}, {9, 19}, {10, 18}, {12, 16}, {13, 15}} {{1, 5}, {2, 19}, {4, 8}, {7, 11}, {10, 14}, {13, 17}, {16, 20}} {{0, 7}, {1, 15}, {3, 10}, {4, 18}, {6, 13}, {9, 16}, {12, 19}} {{0, 11}, {2, 12}, {3, 14}, {5, 15}, {6, 17}, {8, 18}, {9, 20}} {{1, 14}, {2, 10}, {4, 17}, {5, 13}, {7, 20}, {8, 16}, {11, 19}} {{0, 8}, {2, 15}, {3, 11}, {5, 18}, {6, 14}, {9, 17}, {12, 20}} {{0, 10}, {1, 12}, {3, 13}, {4, 15}, {6, 16}, {7, 18}, {9, 19}} {{0, 13}, {1, 12}, {3, 10}, {4, 9}, {6, 7}, {15, 19}, {16, 18}} {{0, 8}, {2, 6}, {3, 5}, {9, 20}, {11, 18}, {12, 17}, {14, 15}} {{0, 14}, {2, 12}, {3, 11}, {5, 9}, {6, 8}, {15, 20}, {17, 18}} {{1, 14}, {2, 13}, {4, 11}, {5, 10}, {7, 8}, {16, 20}, {17, 19}} {{1, 8}, {2, 7}, {4, 5}, {10, 20}, {11, 19}, {13, 17}, {14, 16}} {{0, 10}, {1, 9}, {3, 7}, {4, 6}, {12, 19}, {13, 18}, {15, 16}} {{1, 8}, {2, 16}, {4, 11}, {5, 19}, {7, 14}, {10, 17}, {13, 20}} {{1, 11}, {2, 13}, {4, 14}, {5, 16}, {7, 17}, {8, 19}, {10, 20}} {{0, 11}, {2, 9}, {3, 8}, {5, 6}, {12, 20}, {14, 18}, {15, 17}} {{1, 11}, {2, 10}, {4, 8}, {5, 7}, {13, 20}, {14, 19}, {16, 17}}

45

46

Aiso Heinze and Mikhail Klin Table 9. (Continued)

0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

{0, 1, 2} {1, 3, 6} {2, 4, 5} {2, 7, 14} {1, 8, 14} {1, 9, 15} {5, 8, 10} {0, 11, 12} {5, 6, 13} {0, 8, 26} {0, 6, 7} {2, 13, 16} {2, 3, 26} {2, 17, 27} {1, 18, 27} {1, 19, 28} {5, 18, 20} {3, 12, 16} {3, 11, 21} {1, 4, 22} {4, 12, 17} {12, 14, 23} {11, 14, 24} {5, 15, 25} {4, 6, 35} {3, 7, 15} {4, 8, 16} {0, 21, 23} {0, 16, 24} {6, 16, 25} {6, 9, 26} {2, 10, 35} {0, 18, 38} {0, 15, 17} {6, 14, 17} {2, 11, 25} {2, 9, 38} {2, 29, 31} {1, 30, 31} {5, 30, 32}

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79.

{5, 9, 12} {9, 11, 22} {4, 7, 21} {9, 16, 21} {7, 10, 12} {7, 13, 22} {10, 17, 24} {1, 12, 20} {1, 13, 33} {0, 5, 34} {4, 11, 18} {13, 17, 21} {4, 26, 33} {12, 13, 29} {10, 22, 26} {12, 27, 36} {11, 27, 34} {5, 28, 37} {13, 15, 35} {7, 8, 27} {7, 9, 28} {10, 16, 18} {3, 22, 23} {4, 24, 29} {1, 10, 21} {14, 21, 36} {14, 16, 34} {15, 16, 37} {14, 18, 26} {15, 19, 26} {8, 20, 35} {3, 8, 38} {4, 15, 41} {3, 17, 28} {8, 11, 13} {0, 22, 36} {6, 11, 37} {6, 19, 38} {2, 20, 41} {0, 30, 39}

Lines of S 80. {0, 28, 29} 81. {6, 27, 29} 82. {2, 23, 37} 83. {2, 19, 39} 84. {12, 19, 35} 85. {11, 19, 33} 86. {5, 19, 21} 87. {17, 22, 25} 88. {17, 20, 34} 89. {16, 19, 22} 90. {7, 20, 24} 91. {8, 23, 25} 92. {7, 25, 33} 93. {1, 24, 32} 94. {1, 25, 40} 95. {5, 14, 40} 96. {3, 5, 24} 97. {13, 18, 23} 98. {12, 26, 32} 99. {13, 26, 40} 100. {0, 35, 40} 101. {4, 23, 30} 102. {21, 25, 29} 103. {20, 21, 26} 104. {3, 34, 35} 105. {4, 38, 40} 106. {12, 25, 39} 107. {20, 22, 38} 108. {12, 31, 41} 109. {11, 31, 40} 110. {23, 24, 27} 111. {25, 28, 35} 112. {8, 17, 31} 113. {7, 18, 31} 114. {7, 19, 30} 115. {16, 20, 30} 116. {9, 23, 33} 117. {10, 29, 34} 118. {10, 11, 30} 119. {6, 10, 41}

120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159.

{13, 24, 39} {21, 27, 41} {16, 27, 40} {16, 28, 32} {26, 27, 30} {26, 28, 31} {18, 32, 35} {13, 28, 41} {9, 17, 30} {3, 33, 36} {4, 34, 39} {14, 22, 41} {11, 15, 32} {14, 15, 29} {14, 30, 38} {15, 31, 38} {8, 32, 41} {3, 18, 39} {4, 28, 36} {3, 29, 30} {0, 33, 41} {6, 23, 32} {6, 31, 39} {2, 32, 36} {21, 31, 35} {22, 29, 37} {17, 32, 40} {5, 22, 31} {18, 25, 36} {17, 33, 37} {16, 31, 33} {7, 32, 34} {8, 36, 37} {7, 37, 40} {1, 34, 37} {5, 27, 33} {9, 24, 35} {24, 26, 37} {25, 26, 34} {14, 33, 35}

160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195.

{13, 30, 36} {12, 37, 38} {13, 34, 38} {21, 37, 39} {10, 33, 38} {9, 34, 41} {22, 32, 39} {24, 31, 36} {23, 31, 34} {30, 35, 37} {8, 9, 39} {8, 19, 29} {19, 23, 40} {20, 29, 40} {10, 23, 28} {10, 15, 36} {24, 25, 38} {21, 32, 38} {25, 30, 41} {17, 18, 19} {11, 20, 28} {9, 36, 40} {10, 39, 40} {6, 20, 36} {22, 27, 35} {15, 27, 39} {27, 28, 38} {18, 37, 41} {9, 18, 29} {3, 40, 41} {15, 20, 23} {14, 28, 39} {29, 32, 33} {19, 24, 41} {20, 33, 39} {19, 34, 36}

The graph C(F, F , F ) is depicted in Fig. 20 on page 48. Its automorphism group has order 3! · 43 · 2. We are now interested in the group G0,1,12 . This group ﬁxes the points 0, 1, and therefore also the point 2. By this reason G0,1,12 ≤ Aut(C(F, F , F )). On the other hand, we clearly have

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47

Fig. 19. Graph C(F, F , F )

G0,1,12 ≤ Aut(C(F, F , F )). We now need to determine information about the intersection of the two groups Aut(C(F, F , F )) and Aut(C(F, F , F )). This may be done by routine simultaneous inspection of both ﬁgures. First we observe that all vertices of the cycle of length 21 remain in their place. This immediately implies that all vertices in the second graph also remain on their place. In other words, Aut(C(F, F , F )) ∩ Aut(C(F, F , F )) is a group of order 1 and therefore we obtain |G0,1,12 | = 1. Summing up our reasoning, we formulate the following lemma which is essential for our description of the group G. Lemma 5. Let F, F , F be the points of the transversal design S = T D(3, 2p) as they were deﬁned above. Let F be a vertex which belongs to the largest connectivity component of the graph M ({F, F , F }). Then for the group G = Aut(S) we have GF,F ,F = {e}.

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Fig. 20. Graph C(F, F , F )

Note that in this case {F, F , F } is called a base of Aut(S). We are now ready to prove Proposition 7. G = G, where G = Aut(S). Proof. We consider the group G = Aut(S) in its action on a set P. Let F, F , F be as they were deﬁned in the previous lemma. We know that G ≤ G and {F, F , F } forms a base of G. Using, where it is convenient, orbits of G instead of orbits of G, we obtain: |G| = 6p · |GF | = 6p · |(F )GF | · |GF,F | = 6p · |(F )GF | · |GF,F | = 6p · 4p · |GF,F | = 6p · 4p · |(F )GF,F | · |GF,F ,F | = 24p2 · |(F )GF,F | · 1 = 24p2 · p · 1 = 24p3 . Taking into account that G ≤ G and |G| = 24p3 , we get that G = G = Aut(S). Recall a simple well-known result from group theory (see, e.g. [85], Theorem 4.19): Proposition 8. If p is an odd prime, then every group of order 2p is either cyclic Z2p or dihedral Dp .

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49

Now we can easily see that Aut(Z2p ) ∼ = Z∗2p ∼ = Z∗p is a cyclic group of order p − 1. A more interesting exercise is to ﬁnd Aut(Dp ). Here we obtain a group of order p · (p − 1), see for details [84]. Thus, we have that |Aut(SRG(Z2p ))| = 3! · (2p)2 · (p − 1) = 24p2 (p − 1) and |Aut(SRG(Dp ))| = 3! · (2p)2 · p(p − 1) = 24p3 (p − 1). Finally, with the Proposition 7 we have proven the following proposition: Proposition 9. Our transversal design S is not isomorphic to a transversal design coming from a group. Our goal is now to point out a translation group H in G = Aut(S). It is clear that H should be a group of order 4p2 , such that each nonidentity element of H does not ﬁx any element of L. For this purpose it is more convenient to work with the action of G on a small set of 3p points. Recall that in terms of this action each element of L is a copy of a cycle C3p of length 3p. Such a cycle can be ﬁxed by permutations only from D3p , these permutations (non-identity) have at most one ﬁxed point on the vertices of Z3p . The group G has a subgroup (Dp × Dp × Dp )pos of order 12 · (2p)3 acting intransitively on Z3p . Let us take any involution i from the third copy of Dp and let us consider a group H = (Dp × Dp × i)pos which has order 1 2 ∼ 2 · 2p · 2p · 2 = 4p . Clearly, as an abstract group we have that H = Dp × Dp . It is easy to check that a non-identity element x of H ﬁxes at least two distinct elements of Z3p . Thus, such an element x does not belong to D3p . Therefore, H is a desired translation subgroup of G, that is a subgroup of G which acts regularly on the 4p2 -element set L. 9.5 Analyzing the Results We can now claim that the desired result is obtained: Theorem 2. Let p be a prime, p ≡ 3 (mod 4) and Sp = S be a structure as described above. Then (a) S is a (resolvable) translation design T D(3, 2p). (b) G = Aut(S) ∼ = (S3 Dp )pos is a transitive group of order 24p3 . (c) G as a transitive group of degree 4p2 in its action on the set L has a regular subgroup H ∼ = (Dp )2 of order and degree 4p2 . (d) The dual structure to S is a 3-net which is not coming from a group of order 2p. The result of Theorem 2 may be reformulated in terms of partial diﬀerence sets, that is there exists a “non-standard” Latin square partial diﬀerence set of size k = 3(2p − 1) over a group (Dp )2 of order 4p2 , where p ≡ 3 (mod 4) is a prime.

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Aiso Heinze and Mikhail Klin

First of all, we recall that a standard partial diﬀerence set Xs corresponds to the SRG(Dp ). Here Xs = {(g, e), (e, g), (g, g −1 )|g ∈ Dp , g = e}, where (x, y) is a generic notation for an arbitrary element from the direct square (Dp )2 of two copies of the dihedral group Dp . We obtain also the existence of a non-standard diﬀerence set Xn analyzing the Latin square graph Γ which exists according to Theorem 2. For example, in case of p = 7 we have the following generators for H = (D7 )2 : a = (0, 3, 6, 9, 12, 15, 18), b = (1, 4, 7, 10, 13, 16, 19),

c = (0, 3)(6, 18)(9, 15)(2, 5)(8, 20)(11, 17), d = (1, 4)(7, 19)(10, 16)(2, 5)(8, 20)(11, 17).

We now have to construct our design S = (P, L) and to ﬁnd a canonical set of 3(2 · 7 − 1) = 39 permutations from H which move a subgraph F to all its possible copies in P. As such we obtain Xn = {a, a2 , a3 , a4 , a5 , a6 , d, ad, a2 d, a3 d, a4 d, a5 d, a6 d, b, b2 , b3 , b4 , b5 , b6 , c, bc, b2 c, b3 c, b4 c, b5 c, b6 c, ab, a2 b2 , a3 b3 , a4 b4 , a5 b5 , a6 b6 , acd, a2 bcd, a3 b2 cd, a4 b3 cd, a5 b4 cd, a6 b5 cd, b6 cd}. In principle, as soon as Xn is found, we can conﬁrm that Xn indeed provides a partial diﬀerence set by direct computations in the group ring over H, see also Sect. 10. We hope that the reader will be able to generalize our construction of Xn from p = 7 to the general case considered in this text. It now remains to describe a Cayley table of the quasigroup Q2p , which is implied by the existence of a transversal design S = Sp . For this purpose we identify the element set of Q2p by that of Dp . In this context it is convenient to consider Dp in its canonical action of degree p on the set Zp = {0, 1, . . . , p−1}. Recall that half of the permutations from Dp are even and half are odd. Let also a = (0, 1, . . . , p − 1) be a canonical generator of Zp . We deﬁne Q2p as follows (here ◦ means the new binary operation in Q2p , while x · y, or more precisely xy means the usual multiplication in Dp ): xya, if x and y are odd, (∗)x ◦ y = xy, otherwise. The proof of the fact that Q2p is exactly deﬁned by (∗) is omitted, though it may be obtained using the methodology described in this paper and with the aid of the partial diﬀerence set Xn presented above. Finally, when our goal was achieved we were surprised by the simple rule (∗), which was obtained. A search in the literature, see details in Sect. 10, allowed us to understand that the quasigroup Q2p is known quite well in the quasigroup theory. We claim that we have succeeded in establishing new properties of Q2p , and more important, to design a technology that “automatically” brings us to the desired result.

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51

10 Conclusion In this section we present various facts which did not ﬁt naturally into the main part of this text, but which are important enough to be mentioned in the paper. 10.1 Partial Diﬀerence Sets and S-rings Schur rings (brieﬂy S-rings) provide an alternative way for considering Latin square graphs which admit a regular group of automorphisms. We refer, e.g., to [102] and to [73] for basic facts about S-rings and their relations to Cayley graphs. The following proposition which can be easily proved (cf. [96]), provides a simple and eﬃcient criterion. (Note that Sprague attributes its formulation to W. McWorter.) Proposition 10. Let K be a group of order n2 . Then the following conditions are equivalent: (1) There exists a Latin square graph Γ over a suitable quasigroup Q of order n such that K is a regular subgroup of T (Q). (2) The group K contains three subgroups X1 , X2 , X3 of order n, such that any two of these subgroups have an intersection of size 1. Remark. In this case we have X = X1 ∪ X2 ∪ X3 \ {e}, where e is the identity element of K. X is the connection set of a Cayley graph over K which is isomorphic to Γ . Such a connection set is called a partial diﬀerence set over K. Note that the advantage of this criterion is that it provides the most natural way to detect a desired partial diﬀerence set in a suitable group K. However, it is not universal, because it may happen that T (Q) does not contain a regular subgroup, while Σ(Q) does. 10.2 Pseudo-geometrical Graphs Proposition 2 in Sect. 3.2 is, in fact, a particular case of a more general question which was considered by Bose for partial geometries and by Bruck for a particular class of partial geometries, namely, k-nets. Therefore, arguments which provide the bound n > 23 are of a general nature. It may happen that for our case of 3-nets a better bound may be found. Table 10 shows how drastically the number p(n) of pseudo-Latin square graphs on n2 vertices increases for small n in comparison with the number m(n) of main classes (the value of p(6) is borrowed from [95]).

52

Aiso Heinze and Mikhail Klin Table 10. Number of main classes and pseudo Latin square graphs

n m(n) p(n)

4 2 2

5 2 15

6 12 32548

10.3 The Group Case We believe that Corollary 1 in Sect. 3.4 was ﬁrst formulated and proved in [90], though in diﬀerent terms. Theorem 1 was formulated in [45] but without proof (the authors at that time were not aware of [90]). A brief proof of both claims is given in [32]. Certain variants of propositions, which are equivalent to Corollary 1 or treat some of its particular cases, appear in the literature sporadically. It seems that some of the corresponding authors were also not aware of [90]. Here we refer to [86] as one of the earliest alternatives to [90]. 10.4 Hamming Graphs and Latin Squares We regard the model of a Latin square as a coclique in the Hamming graph, which was presented in Sect. 4 on a level of a few examples, as a methodological innovation. Note that implicitly similar links were mentioned in the literature, see e.g. [16, 11, 93]. Unfortunately, this methodological approach at the present time does not seem to be very practical, when we face a problem of the constructive enumeration of Latin squares of order n for relatively large values of n. 10.5 Loops and Permutations In understanding a permutation of n elements as a bijection of a set [1, n] onto itself, we say that two permutations are discordant if no symbol has the same image under both permutations. A complete set of discordant permutations of the set [1, n] consists of n mutually discordant permutations. Such a set may be naturally associated to the rows of a Latin square Q. A corresponding representation of Q is called by D´enes and Keedwell [35] a Cayley-Sch¨ onhardt representation (or CS-representation) with credits to [28] and [90]. We were using CS-representations in Sect. 4.6 in order to distinguish isomorphism classes of loops. Interesting applications of CS-representations for the evaluation of the number of isomorphically distinct 1-factorizations of the complete directed graph may be found in [36]. 10.6 Order 5 Euler was the ﬁrst who, using an exhaustive process, completed the enumeration of all 56 normalized Latin squares of order 5 [37]. Cayley used what is

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53

now called the CS-representation of Latin squares in order to reach a more reﬁned classiﬁcation [28]. Besides obvious interest to Latin squares of order 5 an expert meets many other attractive reasons of investigation. On the one hand, order 5 is the smallest case where we face the occurrence of the existence of non-geometrical strongly regular graphs with the parameters of a graph SRG(Q), see again Table 10. On the other hand, in order 5 we meet for the ﬁrst time proper loops, as they were presented in Sect. 4.6. In fact, these loops provided an attractive lead for generations of mathematicians. A partial list of important references includes such names as: • • • • • • • •

•

A. Albert – enumeration of all loops of order 5, [2]; Bruck – discussion in [23] of a loop of order 5 with a transitive automorphism group; D. A. Norton, S. K. Stein – discussion of two examples of quasigroups of order 5 in the context of a developed concept of a cycle in an arbitrary Latin square, [75]; R. Artzy – demonstration of a loop (L1 from Sect. 4.6) as an example of a crossed-inverse loop that yields a principal isotope which is not crossedinverse [3]; R. Burn – an expository paper, in which the same loop L1 serves as an example of a loop which does not satisfy the Lagrange theorem, see [27]; A. D. Keedwell – presents a symmetric idempotent quasigroup of order 5 which provides a decomposition of the graph K5∗ into quadrangles that satisfy certain prescribed properties, [57]; A. V. Kuznetsov, E. A. Kuznetsov – consideration of a quasigroup of order 5 in the context of constructions of projective planes, [63]; T. Bier, P. K. Chua – various examples of small loops (starting from order 5) that serve as a motivation for the consideration of a new (at that time) class of numerical invariants of SRG’s, called totient numbers, which were introduced as a generalization of invariants of nets considered by Bruck in his classical paper [22], see [18]; R. Peeters – characterization of the Paley graph P (25) in totality of 15 non-isomorphic srg(25, 12, 6, 6) with the aid of a 5-rank linear combination of matrices A and I, [78].

10.7 Order 6 The investigation of Latin squares and loops of order 6 is a topic that requires the special attention of experts in the history of combinatorics. One of its subdivisions is a proof of the fact that two orthogonal Latin squares of order 6 do not exist. G. Tarry [98] was the ﬁrst who established 17 types in order 6 and obtained the value R6 = 9408.

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A more detailed classiﬁcation of Latin squares of order 6 was given by Sch¨ onhardt in his classical paper [90]. He provides a reasonably detailed bibliography, giving credit to many predecessors, in particular to Tarry. Unfortunately, for a few decades this classical paper was unknown to many researchers. It seems that R. A. Fisher and F. Yates were not aware of Tarry’s results. The motivation for them to enumerate all Latin squares of order 6 came from a paper of S. M. Jacob (1930) who managed to ﬁnd just 8192 normalized squares of order 6. Note that Fisher and Yates found in [42] the correct numbers 12, 17 and 9408 corresponding to row 6 in Table 1 of Sect. 4. Following [38], we distinguish constructive and analytical enumeration of combinatorial objects. In this context almost all of the cited results are referred to the constructive enumeration. P. A. MacMahon presented in [66] a general method for an analytical enumeration of Latin squares of arbitrary order n based on the use of diﬀerential operators. MacMahon demonstrated his method on n = 4, saying that it became impractical for higher orders. Note that MacMahon, who has the merited reputation as one of the founders of combinatorial analysis, made a mistake (working 150 years later than Euler) in the number R5 as the result of his constructive search (as it was reported in [42]). An impressive resurrection of MacMahon’s method was undertaken by P. N. Saxena. Using and developing the analytical techniques of the great predecessor Saxena found in [88], the correct value for R6 . Moreover, in [89] the same method was used to provide a correct number R7 = 1694200. We refer to [4] who speaks about 22 isotopy classes of order 6, and to [26] for the correct number (equal to 109) of loops of order 6. A detailed and helpful catalogue of all quasigroups of order up to 6 was given by Sade in [87]. Independently of [90] the description of the automorphism groups for all 17 types of order 6 is given in [67]. D. Betten in [16] gives a self-contained classiﬁcation of main classes of order 6. He pays special attention to a Latin square that is mostly related to the problem of the existence of a pair of orthogonal squares. This Latin square belongs to the main class introduced in Sect. 4.6. Betten gives a strict description of the square in terms of the 30 edges of the Dodecahedron and the group A5 . An interesting analysis of the 12 main classes of order 6 was done with the aid of a computer in [41] and [60], classifying all mutually orthogonal partitions of these squares. Latin squares of order 6, as well as of other small orders, were used in diverse publications for illustrative purposes. Here we provide just a few of corresponding references. A. Suschkewitsch in [97] was using a kind of local switching of the Cayley table of the symmetric group of order 6 to obtain examples of quasigroups, satisfying certain postulates introduced in his paper.

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E. T. Parker [76] and K. Heinrich [49] considered certain approaches in order to measure how far one may approximate (non-existing) pairs of orthogonal Latin squares of order 6 by their analogues. 10.8 Vertex-transitive Graphs It was proved in [80] that every abstract group is isomorphic to the automorphism group of a suitable Latin square graph. Another paper of Phelps [81] established that for n ≥ 7 there exists a Latin square graph with the automorphism group of order 1 and that the number of such graphs goes to inﬁnity with n. Arguments by Cameron, brieﬂy mentioned in the discussion section of [8], justify the fact that with the growing n almost all Latin square graphs have such a property. The other extremum is related to a question: which Latin square graphs have a transitive automorphism group? Clearly, this is fulﬁlled for the Cayley table of groups. There exists, however, many examples of proper loops, for which the Latin square graph has a transitive automorphism group, see Sect. 13 of [14]. As we mentioned in the Introduction, the same mathematical objects and propositions concerning Latin squares appear in diﬀerent fashion and terminology in the statistical design of experiments, graph theory, general algebra, group theory, geometry and algebraic combinatorics. As a result, experts in diﬀerent areas are sometimes not aware of the achievements of others. We hope that this paper will help build new bridges between all the mentioned areas. 10.9 Q6 The loop Q6 in fact was attracting special attention among many experts, especially due to the properties of the group Aut(SRG(Q6 )) and some of its subgroups. This loop was also considered in [8]. Among other recent interesting discussions of the properties of Q6 and its associated groups we mention [74] and [62]. However, a very signiﬁcant group-theoretical analysis of the net corresponding to Q6 was already presented by Sprague in the article [96], a publication that was inspired by [55]. (In fact, the paper [96] was written in 1978– 1979 and was initially entitled “Nets with Singer group ﬁxing each parallel class”.) Sprague denotes Q6 by O, and shows that the net corresponding to it contains a vertex transitive group of order 36 isomorphic to S3 × S3 . He also presents a corresponding partial diﬀerence set in S3 × S3 . Nevertheless, the complete group Aut(SRG(Q6 )) is not investigated. It is also mentioned that Z6 , S3 , and O are the only loops which imply Latin square graphs corresponding to translation nets of order 6. Though Sprague’s paper was cited e.g., in [12, 56, 48], it seems as though it has not received the credit it deserves.

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10.10 Q2p Loops, which are isomorphic to all their loop isotopes, are usually called G-loops, or loops with the isotopy-isomorphy property. Each group is a G-loop, but there are many proper G-loops. First stages in the development of the theory of G-loops may be attributed, in particular, to [25] and [15]. Due to a pioneering result of E. Wilson [103], which essentially used [26] in order to prove that a prescribed loop L is a G-loop, it is enough to concentrate on two special kinds of isotopes. An essential breakthrough in the theory of G-loops was done by R. L. Wilson Jr. in his Ph.D thesis (1969), written at the University of Wisconsin under direction of Hans Schneider, and two subsequent papers. In [104], using short combinatorial arguments, and based again on [26], it was proved that a loop of prime order is a G-loop if and only if it is a cyclic group. In the next paper [105] it is proved that for each even number 2n there exist a loop of order 2n (we prefer to denote it by Q2n , it is commonly called the Wilson loop) which turns out to be a proper G-loop. In fact, for a prime p the Wilson loop of order 2p strictly coincides with the loop Q2p rediscovered by us in this paper as a theoretical generalization of the results of our computer algebra experimentation. This class of Wilson loops has been considered in the literature many times, see e.g., [46, 47]. There is a speciﬁc variety of G-loops which is called the conjugacy closed loops, brieﬂy CC-loops. This class was introduced in [47] as loops, in which the right and left multiplications are closed under conjugations. An equivalent deﬁnition is suggested in [62], namely a loop should satisfy two identities RCC: z(yx) = ((zy)/z)(zx)

and

LCC: (xy)z = (xz)(z \ (yz)).

In this paper Kunen characterizes the loops Q2p : If p is an odd prime, then there are exactly three CC-loops of order 2p, namely two groups and the loop Q2p . Our new input into the theory of loops Q2p is the precise description of the automorphism groups of Γ = SRG(Q2p ), as well as detecting the regular subgroups in Aut(Γ ). Kunen presents in another paper [61] a wide and deep investigation of various permutation groups related to G-loops. We have no doubt that a careful analysis of the results in [61] may suggest an alternative way to describe the groups Aut(Γ ) for Q2p . 10.11 Erich Sch¨ onhardt The article [90] by E. Sch¨ onhardt from T¨ ubingen is certainly an outstanding event in the history of combinatorial mathematics in the 20th century. We have already discussed a few signiﬁcant achievements presented in it. Contrary to many future experts in the theory of Latin squares, Sch¨onhardt was widely knowledgeable of the results of his predecessors; here is (an incomplete) list of names getting credit in [90]: Euler, Cayley, E. Netto, M. Frolov,

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MacMahon, Tarry, E. Skolem, E. Coursat, A. Speiser, H. Brandt, W. Burnside, G. Frobenius, G. A. Miller and others (the ordering is according to the ﬁrst appearance of the name in the text). The paper consists of three chapters. The ﬁrst chapter deals with Latin squares and various groups associated to it. The second chapter provides a systematical investigation of various relations between Latin squares, quasigroups and associated groups. The last chapter presents a complete catalogue of Latin squares of order 5 and 6 together with certain groups associated to them. Unfortunately, a reader (even one who was born and raised in Germany, as one of us) faces essential diﬃculties in the understanding parts of the text due to the non-standard terminology used by Sch¨ onhardt. As it was mentioned, generations of researchers, not familiar with [90], were forced to create their own terminology which has remained very diverse for many decades. We are sure that a careful translation of [90] will still provide a brilliant source of fresh scientiﬁc information. 10.12 Article by Brendan McKay et al. The article [69] is deﬁnitely a landmark in the enumeration theory of Latin squares, and probably the most signiﬁcant event in its history since the time of Sch¨ onhardt. For many decades this mathematical theory suﬀered from excessive and idiosyncratic terminology and notation, that come from diverse traditions of many adjacent mathematical domains (from loop theory to combinatorial designs). Based mostly on the practice of Sade, the authors in [69] attempt to unify the notation. This standardization will hopefully inﬂuence researchers in the future. Our Table 1 in Sect. 4 was strongly inﬂuenced by the acquaintance with a preliminary version of the paper by McKay et al. We were also trying to make our bibliography complementary to their list of 55 references. 10.13 A Few Books Below we list a few books which may be helpful to the reader. The book [65] acquaints with Latin squares, strongly regular graphs, nets and association schemes. A detailed consideration of transversal designs and nets may be found in [17]. The classical book [34] has served for many decades as a comprehensive source on Latin squares, it has a very detailed and annotated bibliography. The monograph [15] is one of the ﬁrst attempts to address the foundations of loops and quasigroups. (One of the authors, who was personally familiar with V. D. Belousov, will forever recall his fascinating features of an algebraist and a person.)

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A more modern text [79] provides a very friendly introduction to loops and quasigroups, it is ﬁlled with examples and helpful exercises. The monograph [93] contains a helpful chapter dealing with groups and quasigroups, considering, in particular, loops, nets and isotopy. The book [10] provides comprehensive information about applications of association schemes in statistics, in particular those schemes which are related to Latin squares. Finally, we highly recommend one more introductory text, namely [64]. It covers many topics, in particular, some links of Latin squares with groups and graphs. 10.14 More References As was already mentioned in Sect. 4.5, the case of strongly regular graphs with the parameter set (v, k, λ, μ) = (25, 12, 5, 6) is the smallest case, where we face signiﬁcant diﬃculties in developing complete constructive enumeration of all objects. This is why this problem became one of the leads attracting for investigators. In fact, four teams of researchers attacked this problem more or less at the same time: Shrikhande and Bhat [92]; Arlasarov, Leman and Rosenfeld ([83, 5], see also [101]); Paulus [77]; and ﬁnally Corneil and Mathon [31]. The latter paper provides the most complete bibliography and many interesting details about all 15 graphs. In particular, the order and the generators of the groups of the graphs on 25 vertices, which appear in Sect. 4, were already presented in [31]. The automorphism groups of all 3-nets on 25 points are presented on the homepage of Moorhouse [71]. During the last few years a number of very interesting strong results related to Latin squares were obtained by Ian M. Wanless, a former student of McKay. We refer here e.g. to [70] and [100]. There are a few attempts in the literature documenting the eﬀorts to ﬁnd a closed formula for the number of distinct Latin squares of order n (see Sect. 10.7 as well as the discussion in [69]). In this context we attract the reader’s attention to the paper [54], which seems to have gone unnoticed by researchers. Here the author suggests an approach that reduces the problem to the computation of some structure constants in a certain algebra of double cosets of the symmetric group of order n2 with respect to its intransitive subgroup. No attempts are known to investigate how practical such an approach may be for small values of n. Section 2.2 of the Thesis [29] discusses helpful isotopy invariants of loops that are formulated in terms of corresponding association schemes. A number of publications were devoted to the investigation of Latin squares with high symmetry regarded in diverse senses. The articles [19] and [33] serve as nice patterns of such investigations. For inﬁnite quasigroups and loops that are strictly related to the investigations of 3-webs in classical geometry, see the recent survey [1].

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And for description of the automorphism groups of dihedral groups see besides [84], paper [99] and references in it. 10.15 The Presented Project The project described in this paper has been ongoing since 2001. Its ﬁrst version was presented at the AMS meeting #991 at Chapel Hill (October 24–25, 2003), see [58]. Since that time we discovered how Q6 may be extended to an inﬁnite series Q2p . A preliminary version of this result was announced in [51].

Acknowledgments Conceptually, this project goes back to the Moscow school, in particular to the collaboration of M. K. with I. A. Faradˇzev, A. V. Ivanov, and especially, with the late Ja. Ju. Gol’fand. More recent collaborations with E. van Dam, M. Muzychuk and A. Woldar have been very helpful. We also acknowledge R. Bailey, D. Keedwell, N. Kriger, R. Mathon, B. McKay, W. Myrvold, Ch. Praeger, S. Reichard, H. Schneider, E. Spence, A. Sprague, and R. L. Wilson for valuable assistance, information and suggestions. This project was initiated in August 2001, when M. K. was visiting Alex Rosa at McMaster University, Hamilton, Ontario. Long discussions with Alex on various issues related to Latin squares were very productive and stimulating. The main part of this contribution was prepared when A. H. was assistant at the Institute of Mathematics, Universit¨ at Augsburg, Germany. At that time M. K. was partially supported by Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. Finally, we would like to thank Ted Eisenberg for his comments on the presentation of this paper.

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80. K. T. Phelps, Latin square graphs and their automorphism groups, Ars Combin., 7 (1979), 273–299. 81. K. T. Phelps, Automorphism free Latin square graphs, Discrete Math., 31 (1980), 193–200. 82. D. A. Robinson, The Bryant-Schneider group of a loop, Ann. Soc. Sci. Bruxelles, S´er. I, 94 (1980), 69–81. 83. M. Z. Rosenfeld, On the construction and properties of some families of strongly regular graphs, Uspekhi Mat. Nauk, 28 (1973), 197–198 (Russian). 84. F. Rotmaler, Automorphism groups of dihedral groups, Ukr. Math. J., 29 (1977), 162–167 (Translation from Russian). 85. J. J. Rotman, An Introduction to the Theory of Groups, 4th edn. Graduate Texts in Mathematics, Vol. 148, Springer, New York, 1995. 86. A. Sade, Quasigroupes isotopes. Autotopies d’un groupe, Ann. Soc. Sci. Bruxelles Sir. III, 81 (1967), 231–239. 87. A. Sade, Morphisms de quasigroups table, Revista Fac. Sciencias de Lisboa, 13 (1971), 149–172. 88. P. N. Saxena, A simpliﬁed method of enumerating Latin squares by MacMahon’s diﬀerential operators. Part I. The 6×6 Latin squares, J. Indian Soc. Argicultural Stat., 2 (1950), 161–188. 89. P. N. Saxena, A simpliﬁed method of enumerating Latin squares by MacMahon’s diﬀerential operators. Part II. The 7 × 7 Latin squares, J. Indian Soc. Agricultural Statist., 3 (1951), 24–79. ¨ 90. E. Sch¨ onhardt, Uber lateinische Quadrate und Unionen, J. Reine Angewandte Math., 163 (1930), 183–230. 91. S. S. Shrikhande, The uniqueness of the L2 -association scheme, Ann. Math. Statist., 30 (1959), 781–798. 92. S. S. Shrikhande and V. N. Bhat, Graphs derivable from L3 (5) graphs, Sankhy¯ a Ser. A, 33 (1971), 315–350. 93. J. D. H. Smith and A. B. Romanovska, Post-Modern Algebra, Wiley, New York, 1999. 94. L. H. Soicher, The GRAPE Package for GAP. http://www.maths.qmul. ac.uk/∼leonard/grape/. 95. E. Spence, Strongly Regular Graphs on at Most 64 Vertices. http://www. maths.gla.ac.uk/∼es/srgraphs.html. 96. A. P. Sprague, Translation nets, Mitt. Math. Semin. Gießen, 157 (1982), 46–68. 97. A. Suschkewitsch, On a generalization of the associative law, Trans. Amer. Math. Soc., 31 (1929), 204–214. 98. G. Tarry, Le probl`eme des 36 oﬃciers, Assoc. Franc. Paris, 29 (1900), 170–203. 99. G. L. Walls, Automorphism groups, Amer. Math. Monthly (6), 93 (1986), 459–462. 100. I. M. Wanless, About Latin squares based on cyclotomic orthomorphisms, Electron. J. Combin., 12 (2005), R22. 101. B. Weisfeiler (ed.) On Construction and Identiﬁcation of Graphs. With contributions by A. Lehman, G. M. Adelson-Velsky, V. Arlazarov, I.

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Siamese Combinatorial Objects via Computer Algebra Experimentation Mikhail Klin1 , Sven Reichard2 , and Andrew Woldar3 1 2

3

Ben-Gurion University of the Negev, Beer Sheva 84105, Israel. [email protected] University of Western Australia, Crawley 6009, Western Australia. [email protected] Villanova University, Villanova, PA 19085, USA. [email protected]

Summary. Following Kharaghani and Torabi [On a decomposition of complete graphs, Graphs Comb., 19 (2003), 519–526], we introduce new concepts of Siamese color graph, Siamese association scheme and Siamese Steiner design. With the aid of a computer, we determine all Siamese objects on 15 points, as well as hundreds on 40 points. As a generalization of accumulated observations, an inﬁnite series of Siamese association schemes related to certain imprimitive actions of the groups PSL(2, q 2 ) is outlined. Special attention is paid to the spirit of computer-aided activity, namely to algorithms, technical data, successful ad hoc tricks, and computer-free interpretations of obtained results.

Key words: Color graph, Coherent conﬁguration, Association scheme, Distance regular graph, Strongly regular graph, Generalized quadrangle, Spread, Steiner system, Siamese color graph, Siamese association scheme, Siamese Steiner design, Computer algebra package

1 Introduction The starting point for this project was our acquaintance with the paper [35], in which an inﬁnite series of special color graphs was constructed. Quite early on, we realized how one could naturally axiomatize the properties of these constructed objects in order to obtain, in our terminology, a Siamese association scheme – more generally, a Siamese color graph. Certain mono- and bi-chromatic graphs related to Siamese color graphs form distance regular and strongly regular graphs; in particular, the strongly regular graphs have the same parameters as point graphs of generalized quadrangles. If moreover, these strongly regular graphs are geometric, then one additional structure is implied – a Siamese Steiner design. In such manner, we came about a quite remarkable and hopefully fruitful link between such well studied objects as color graphs, association schemes M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 2,

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and Steiner designs – all termed by us Siamese objects. As a next step, we constructed and investigated many such objects at both the computational and theoretical level. From the early inception of our project, the use of computers was an inalienable part of all our activities. Indeed, it would be fair to say that virtually all theoretical results presented in this paper are generalizations of observations gleaned through one or another computer experiment. A brief account of our discoveries in a form very close to the style of an extended abstract is presented in [39]. Elements of the general theory of Siamese objects will be developed in forthcoming papers, particularly in [40, 41]. In the course of our work over these papers, we realized that there is an excellent opportunity to prepare a text of absolutely diﬀerent genre – one that lies somewhere between the two extreme cases of extended abstract and comprehensive treatise. This is the genre of a tutorial paper, which is presented in the current text. One of our multiple objectives is to provide a brief and reasonably friendly introduction to various concepts from algebraic combinatorics, including coherent conﬁgurations, association schemes, distance regular graphs, strongly regular graphs, etc. It is hoped that this portion of the text will serve not only as an aid to the readers of this paper, but to those who have scientiﬁc interest in any number of accompanying contributions comprising this collection. We are, however, attempting to achieve this objective in conjunction with a second important goal – to introduce the reader to the recently formulated notion of Siamese combinatorial objects, with suﬃcient attention paid to theoretical aspects, striking examples, and relevant links to objects from other areas of combinatorics such as geometry and design theory. The word “algorithmic,” as it appears in the title of the entire collection of papers, clearly dictates what are our remaining goals: • To describe the main methodological features of our vision of computer algebra experimentation, as it applies to combinatorics. • To introduce the reader to extant computer packages such as COCO and GAP. • To share our experience, obtained at the forefront of computational and theoretical exploration. • To challenge the reader to extend, or otherwise generalize, our results. We conclude with a brief outline of the balance of the paper. In Sect. 2, all preliminary information is gathered. Computer packages are discussed in Sect. 3, together with general features of our methodology. Section 4 is devoted to main deﬁnitions and simple theoretical facts related to Siamese objects, while Sect. 5 provides an overview of our activities. (Section 5 may be regarded as a synopsis of material to appear in [40] and [41].) The main scientiﬁc load of the paper occurs in Sects. 6, 7, 8 and 9, where we present all experimental and theoretical results concerning Siamese objects on 15 and 40

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points. We conclude with additional remarks in Sect. 10, including some of purely speculative nature.

2 Preliminaries 2.1 Color Graphs Deﬁnition 1. A color graph Γ is a pair (V, R), where V is a set of vertices and R a set of (non-empty) disjoint binary relations on V such that R∈R R = V 2 . We refer to the elements of R as the colors of Γ , and to the number |R| of its colors as the rank of Γ . In other words, a color graph is an edge-coloring of a complete graph. Note that any function φ deﬁned on V 2 deﬁnes a color graph. Given a color graph Γ , we deﬁne its adjacency matrix to be the v × v matrix A = (aij ) for which aij = t if (xi , xj ) ∈ Rt , Rt ∈ R. Deﬁnition 2. Let Γ = (V, R) and Γ = (V , R ) be color graphs. An isomorphism φ : Γ → Γ is a bijection of V onto V which induces a bijection ψ : R ↔ R of colors. A weak (or color) automorphism is an isomorphism φ : Γ → Γ . If, in addition, the induced map ψ is the identity on R we call φ a (strong) automorphism. We denote by CAut(Γ ) and Aut(Γ ) the groups of weak and strong automorphisms of Γ , respectively. We shall often refer to Aut(Γ ) as the group of Γ , and to CAut(Γ ) as the color group of Γ . 2.2 Coherent Conﬁgurations and Association Schemes A coherent conﬁguration is one of the initial notions which, in principle, we presume to be known (e.g., see [29, 18, 12]). However, to keep our text selfcontained we give its deﬁnition. Deﬁnition 3. A color graph M = (X, R), R = {Ri | i ∈ I}, is a coherent conﬁguration if the following conditions are satisﬁed: (1) The identity relation IdX = {(x, x) | x ∈ X} is a union of suitable relations Ri , i ∈ I , I ⊂ I. (2) For each i ∈ I there exists i ∈ I such that Rit = Ri , where Rit := {(x, y) | (y, x) ∈ Ri }. (3) For any i, j, k ∈ I, the number pkij of elements z ∈ X for which (x, z) ∈ Ri and (z, y) ∈ Rj is constant, provided (x, y) ∈ Rk . The constants pkij appearing in Deﬁnition 3 are called intersection numbers (or structure constants) of M. Given a coherent conﬁguration M = (X, {Ri }) we refer to the relations Ri as basis relations. The graphs Γi = (X, Ri ) will be called basis graphs, whereas

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their adjacency matrices Ai = A(Γi ) will be called basis matrices. This allows us to switch freely between the languages of matrices, relations and graphs. Because a coherent conﬁguration is deﬁned as a particular case of a color graph, all notions deﬁned for color graphs apply to coherent conﬁgurations as a special case. Similarly, we may regard association schemes as a particular class of coherent conﬁgurations (see [18]). For completeness, we give its deﬁnition as well. Deﬁnition 4. A coherent conﬁguration M = (X, {Ri }) is an association scheme if the identity relation IdX is one of the basis relations of M. Typically, we denote this basis relation by R0 . We stress that in our terminology an association scheme is not presumed to be symmetric or commutative, see [1]. In what follows we call the identity relation IdX = R0 of an association scheme deﬁned over X the trivial relation, while all nontrivial relations are called classes. We call a basis graph nontrivial if its arc set is a class. In a primitive association scheme all nontrivial basis graphs are connected; otherwise the scheme is imprimitive. A class of examples of association schemes arises from distance regular graphs: Deﬁnition 5. Let Γ be a connected graph of diameter d. Suppose that for any pair of vertices (x, y), the number of vertices z at distance i from x and at distance j from y depends only on the distance k of x and y. Then Γ is called a distance regular graph (brieﬂy, drg). Given such a graph Γ , we can deﬁne relations Ri , i = 0, . . . , d, with (x, y) ∈ Ri if their distance in Γ is i. Proposition 1. The relations Ri deﬁned above form an association scheme with d classes. Association schemes coming from distance regular graphs are called metric, or sometimes P -polynomial. We will use a number of additional notions, including imprimitive drg, antipodal cover and quotient graph. We refer the reader to [9] for their discussion. Let Γ = (V, E) be a ﬁnite simple graph, that is, undirected, without loops or multiple edges. The order of Γ is v = |V |. The number of neighbors of a vertex x ∈ V is called the valency of x. Recall that Γ is called regular if all its vertices have the same valency k. In such case, we further say that Γ has valency k. Although a strongly regular graph is a particular case of a drg, we prefer to deﬁne this notion in a separate self-contained manner.

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Deﬁnition 6. Let Γ be a regular graph of order v and valency k. Suppose there exist nonnegative integers λ and μ such that all pairs of adjacent vertices in Γ have λ common neighbors, and all pairs of distinct non-adjacent vertices have μ common neighbors. Then Γ is a strongly regular graph (brieﬂy, srg) with the parameters (v, k, λ, μ). (We sometimes refer to Γ as an srg(v, k, λ, μ).) Graphs of the form m ◦ Kn (m disjoint copies of the complete graph Kn ) comprise the class of srg’s with μ = 0. As the complement of an srg is again an srg, we automatically obtain a second class of imprimitive srg’s, namely that consisting of the so-called complete multipartite graphs m ◦ Kn . Example 1. The Petersen graph P is one of the most famous strongly regular graphs. Formally, it can be deﬁned as the complement T (5) of the triangular graph T (5). In other words, vertices are 2-element subsets of a ﬁxed 5-element set, with two vertices adjacent if and only if their corresponding subsets are disjoint. 2.3 Incidence Structures 2.3.1 General Deﬁnitions Deﬁnition 7. Let P and B be sets, and let I ⊆ P × B be a relation, referred to as incidence. Then (P, B, I) is called an incidence structure. Usually, the elements of P are called points, and those of B, blocks. However, sometimes in place of block one uses the more geometric term line. Deﬁnition 8. Let (P, B, I) be an incidence structure. The incidence structure (B, P, I T ) obtained by interchanging blocks and points, and reversing the incidence relation, is called its dual structure. Given an incidence structure, we can naturally deﬁne three graphs related to it. The incidence graph (or Levi graph) is the bipartite graph deﬁned on P ∪ B where a point and a block are adjacent if they are incident. The point graph is deﬁned on P with two points adjacent if they are “collinear” (i.e., incident with a common block). Finally, the block graph is the point graph of the dual structure (i.e., vertices are blocks, and two blocks are adjacent if they are incident to a common point). 2.3.2 Steiner Systems Deﬁnition 9. Let (P, B) be an incidence structure in which each block contains k points and each set of t points is contained in a unique block. Then (P, B) is called a Steiner system S(t, k, v), where v = |P |. If t = 2, k = 3, we speak of a Steiner triple system, or ST S(v). We refer the reader to [31] for a concise treatment of incidence structures, and to [2] for detailed information on Steiner systems.

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Example 2. Consider the 4-dimensional vector space F 4 deﬁned over the ﬁnite ﬁeld F of q elements. Let V and B denote, respectively, the set of all 1-dimensional and 2-dimensional subspaces of F 4 . Deﬁne incidence to be ordinary inclusion. Then the resulting incidence structure (V, B) provides a classical example of a Steiner system S(2, q + 1, q 3 + q 2 + q + 1). We denote it by P G(3, q) and call it projective 3-space over F . Elements of V are called projective points and those of B, projective lines. Note that traditionally P G(3, q) also includes “planes,” which correspond to 3-dimensional subspaces of F 4 . 2.3.3 Generalized Quadrangles Deﬁnition 10. A partial geometry with parameters (K, R, T ) is an incidence structure such that each block (or line) contains K points, each point lies on R lines, each pair of distinct points lies on at most one line, and for each line l and point P not on l, there exist exactly T lines through P that intersect l. Deﬁnition 11. A generalized quadrangle (brieﬂy, GQ) is a partial geometry with T = 1. The pair (s = K − 1, t = R − 1) is called the order of the generalized quadrangle. We will denote a generalized quadrangle of order (s, t) by GQ(s, t). In the case of s = t, we will simply speak of a GQ of order s and write GQ(s). Theorem 1. The point graph of a GQ(s, t) is strongly regular, with parameters v = (s + 1)(st + 1),

k = s(t + 1),

λ = s − 1,

μ = t + 1.

We often refer to the point graph of a GQ(s, t) as a GQ(s, t)-graph. Example 3. A classical example of a GQ(2) is constructed with the aid of a 6element set X, wherein points correspond to 2-element subsets of X and lines to partitions of X into three parts of equal size. Incidence is containment. Fulﬁllment of the axioms of a GQ is an easy exercise for the reader. The point graph of this GQ(2) is the complement graph T (6) of T (6). It is an srg(15, 6, 1, 3). This structure is a classical mathematical object which goes back to J.J. Sylvester [64], who called it the “duad-syntheme geometry.” We refer the reader to [54] for more information on generalized quadrangles. 2.4 Kramer–Mesner Method and Related Issues We now brieﬂy discuss a method which, in principle, allows one to construct all incidence structures with prescribed parameters, which are invariant with respect to a given permutation group (H, Ω). It is usually called the Kramer– Mesner method, due to its formal presentation in [43].

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Table 1. Matrix M = m(i, j) of Example 4 T1 T2

K1 1 0

K2 3 0

K3 6 4

K4 3 8

K5 0 1

Σ 13 13

Below we provide an outline of the algorithm as it applies to the special case of Steiner systems. Thus, we presume we are looking for systems S(t, k, v) which are invariant with respect to (H, Ω), where |Ω| = v. Our treatment follows closely the spirit of [5]. Denote by T the set of all orbits of (H, Ω) in its induced action on the t-element subsets of Ω, and let a = |T |. Similarly, denote by K the set of all orbits of (H, Ω) on the k-element subsets of Ω, and let b = |K|. We now form an a × b matrix M , with rows and columns indexed by the elements of T and K, respectively, in which the entry m(i, j) of M is deﬁned as follows: Let X ∈ Ti be an arbitrary t-element set in the i-th orbit Ti of T . Then m(i, j) = |{Y ∈ Kj | X ⊆ Y }|, where Kj is the j-th orbit of K. It is evident that m(i, j) does not depend on our choice of X ∈ Ti . Suppose we are able to ﬁnd a collection {j1 , j2 , . . . , js } of columns such that s their sum is equal to the all-ones vector of length a. Then, setting B = e=1 Kje , we see that (Ω, B) is evidently an S(t, k, v). Conversely, every such Steiner design which is invariant with respect to (H, Ω) arises in this manner. Example 4. Let us construct an example of an ST S(15) using the Kramer– Mesner method. Consider the symmetric group S6 in its natural action on X = {1, 2, 3, 4, 5, 6}, and let Ω denote the set of all 2-element subsets of X. Because we are looking for an S(2, 3, 15), we have to describe all orbits of (S6 , Ω) on 2- and 3-element subsets of Ω. It is well known that such orbits above are in bijective correspondence with the isomorphism classes of graphs on six vertices having two and three edges, respectively. Thus, we switch to graphical language. Let K6 denote the complete graph with vertex set X. Then the set Ω corresponds to the edge set of K6 , and the entry m(i, j) corresponds to the number of ways one can extend a ﬁxed copy of Ti (as a two-edge subgraph of K6 ) to a graph isomorphic to Kj by adding one new edge. Here, T1 denotes the path of length 2, T2 the graph with two disjoint edges, K1 the triangle, K2 the star, K3 the path of length 3, K4 the graph with two components (a two-path and an edge), and K5 the graph with three disjoint edges. As the reader may easily verify, we obtain the matrix M in Table 1, represented in tabular form. We also include a control sum Σ, which is nothing more than the ordinary column sum of M . (Note that Σ always coincides with |Ω| − t, in our case 15 − 2 = 13.) Now an easy inspection reveals that the ﬁrst and last columns of M induce an ST S(15). In other words, the design will have as its points the 15 edges of the complete graph K6 , and as blocks the 20 triangles of K6 (contributed from

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column 1) plus the 15 1-factors of K6 (contributed from column 5). Thus, we get an STS (15). This design will play an essential role in our presentation. Remark 1. The STS (15) just constructed is a particular case of a so-called graphical design, see [5] for a systematic discussion. 2.5 Double Cosets An important class of association schemes, namely schemes of 2-orbits of transitive permutation groups (equivalently, centralizer algebras of such groups), may be formulated in purely group theoretic terms. Let (G, Ω) be a transitive permutation group, and let H = Gα be the stabilizer in G of a point α ∈ Ω. Let 2-orb(G, Ω) = {R0 , R1 , . . . , Rd } be the set of orbits of the induced action of (G, Ω) on Ω 2 , where R0 , as usual, denotes the 2-orbit with representative (α, α). Then, for each 0 ≤ i ≤ d, the set {g ∈ G | (α, αg ) ∈ Ri } is a so-called double coset of H in G, that is a subset of G of the form Hgi H for a suitable gi ∈ G. It turns out that double cosets are in bijective correspondence with the elements of 2-orb(G, Ω), an observation which was used implicitly throughout the early part of the 20th century. During the last few decades the theory has found numerous applications in computer algebra, e.g., see [18] and [42] for details. We mention that the transitivity assumption is actually not required in the establishment of the correspondence between 2-orbits and double cosets, although it does lead to a marked simpliﬁcation. Example 5. Let us illustrate by way of example the just deﬁned notion of double coset. We take G = (1, 2, 3, 4), (1, 3) = D4 , the dihedral group of order 8 and degree 4. For convenience, we give an explicit listing of its elements: e, (1, 2, 3, 4), (1, 3)(2, 4), (1, 4, 3, 2), (2, 4), (1, 2)(3, 4), (1, 3), (14)(23). Clearly G is a transitive group, with stabilizer H = G1 = {e, (2, 4)}. It is now easy to compute: H = HeH, H(1, 3)H = {(1, 3)(2, 4), (1, 3)}, and H(1, 2, 3, 4)H = {(1, 2, 3, 4), (1, 2)(3, 4), (1, 4, 3, 2), (1, 4)(3, 2)}. Using these three double cosets, the reader can easily see that the corresponding three 2-orbits of D4 are those with representatives (1, 1), (1, 3) and (1, 2), respectively.

3 Computer Algebra Tools 3.1 Computations in Combinatorics The advent of computers in algebraic combinatorics is a relatively recent event. In yesteryear nearly all results were accomplished purely at the theoretical level, but today the role of machine is pervasive, being used to search for

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interesting objects, to determine their combinatorial and algebraic properties, and to generate conjectures. In particular, the creation of symbolic algebra systems has had profound impact on this theoretical-experimental synergy, which nowadays seems indispensable. In addition to algorithms designed to perform both general and speciﬁc tasks, computer algebra systems come with, in many cases, substantial libraries of combinatorial and algebraic data that have been accumulated by generations of mathematicians. We distinguish two kinds of such packages. General purpose systems are designed to deal with a broad range of mathematical problems. Examples are GAP [59] and Maple, which include their own programming languages in order to be extensible. In contrast, specialized packages are designed for one speciﬁc task. Examples here are COCO [17] for the investigation of coherent conﬁgurations, and Discreta for the construction of t-designs. 3.2 COCO COCO (COherent COnﬁgurations) is a collection of programs designed speciﬁcally for the investigation of coherent conﬁgurations. It was developed around 1990 by members of the Moscow Seminar on the Algebraic Theory of Combinatorial Objects, in particular, I. A. Farad˘zev and M. Klin [17]. Originally written in Fortran-4, it has since been ported to C and adjusted for use on personal computers. The next version, which has been slightly modiﬁed and ported to UNIX by A. E. Brouwer, is available from his homepage [8]. In addition to having a modest library of group theoretic data, COCO provides the following facilities: • Inducing Given a permutation group (G, Ω) and a combinatorial structure X, compute the orbit XG and the action of G on XG . • Color Graph Given a permutation group (G, Ω), compute its centralizer algebra V (G, Ω). • Intersection Numbers Given a coherent conﬁguration W , compute its intersection numbers (i.e., structure constants of its corresponding coherent algebra). • Subschemes Given the intersection numbers of a coherent conﬁguration, determine all mergings of its classes which lead to homogeneous subconﬁgurations (fusion association schemes). • Automorphism Groups Given a coherent conﬁguration and a set of mergings, compute the automorphism groups of the resulting fusion schemes. Typically, these ﬁve commands are used in the order speciﬁed. This allows one to determine all association schemes invariant with respect to a given permutation group (G, Ω), as well as their automorphism groups (i.e., the 2-closed overgroups of (G, Ω) in SΩ ).

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Regarding the process of inducing, we mention that orbits are generally easier to compute than coset representatives. Thus, given a (transitive) permutation group (G, G/H) it is always convenient to have a combinatorial structure X for which the action of G on the orbit XG is similar to the action of G on G/H. (In such case, H is the stabilizer in G of X.) This provides a time- and space-eﬃcient way to delineate permutation representations of ﬁnite groups. In general, it may be diﬃcult to verify that H is the full stabilizer of X in G. In contrast, in the case where H is maximal in G one may establish this fact by simply showing that X is invariant under H but not under G. Thus, it is especially easy to ﬁnd representations for primitive permutation actions of G. In the case of imprimitive representations, ad hoc tricks may be helpful. 3.3 GAP Whereas COCO is designed for highly specialized tasks, GAP provides solutions for a wide range of problems in discrete mathematics, with a strong focus on group theory. GAP (Groups, Algorithms, Programs) was originally developed during the 1990’s at the Rheinisch-Westf¨alische Hochschule in Aachen, Germany [59]. More recently, its main development has shifted to St. Andrews, Scotland. The GAP package contains a relatively small kernel written in C, which makes it portable to all current operating systems. This kernel contains basic IO-capabilities, representations of primitive data types such as arbitrary precision integers, ﬁnite permutations and ﬁnite ﬁeld elements, and an interpreter for the GAP language. In the most current versions, this language has been enhanced with object-oriented features such as data encapsulation and runtime polymorphism. What makes GAP so powerful is its vast library of routines and data written in the GAP language. It contains state-of-the-art algorithms for the investigation of permutation groups, matrix groups, rings, ﬁnite ﬁelds, etc. One additional feature is the support of so-called share packages, which are independent routines, typically written by diﬀerent authors, to deal with speciﬁc algebraic or combinatorial objects. 3.4 GRAPE One of the share packages mentioned above is GRAPE (GRaph Algorithms using PErmutation groups) developed by L. Soicher [62]. Its basic premise is that each graph is invariant under some group (possibly the trivial group) acting on its vertices. Such a group is stored together with the ambient graph in order to make the representation more compact, and thus speed up calculations. In fact, many of the algorithms rely on backtracking, and the known group is used to perform isomorph rejection.

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Among the operations available for graphs are global invariants (diameter, girth, independence number, distance-regularity), structural determinants related to subsets of vertices (induced subgraphs, cliques, connected components), and so on. More elaborate algorithms are provided, for example, to enumerate all partial geometries having a prescribed point graph. 3.5 nauty Over the last decade, the standard program for graph isomorphism problems is B. McKay’s nauty (no automorphisms, yes?) [49]. Its name reﬂects the fact that, statistically speaking, a random graph has trivial automorphism group. This fact is used heuristically to ﬁnd invariants which distinguish vertices of the graph; thus the algorithms are especially successful when the automorphism group is small. For a given graph, nauty computes the automorphism group together with a canonical labeling of the vertices. The premise here is that two canonically labeled graphs are isomorphic if and only if they are equal; hence nauty enables a quick isomorphism check (quadratic in the number of vertices). Though the determination of a labeling is not computationally economic, it need only be performed once for each graph under consideration. nauty is supplied with its own language and ﬁle format; however, there also exists an interface to it, provided courtesy of GRAPE. Thus, one need not learn another language in order to beneﬁt from nauty’s most important capabilities, which are consequently available through GAP. 3.6 Computer Algebra Experimentation As has been mentioned several times, in the course of our investigation of Siamese objects we relied quite extensively on the use of computers. Below we distinguish the methodological stages of their use, which, from our own experience, are consistent with the steps taken in the investigation of general combinatorial structures. • Implement existing programs to obtain quick results for relatively small objects. • Determine limitations of existing programs. • Create special task-oriented algorithms to enhance or replace available programs. • Implement new algorithms on known objects as a means of measuring their eﬃcacy. • Search for objects of larger size, as well as establish new properties of known objects. • Interpret and generalize experimental results on a theoretical level.

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3.7 Explanation Versus Interpretation As we have maintained throughout, the discovery of new combinatorial objects, or of unknown properties of known objects, is an immediate goal of computer algebra experimentation. The construction of a new incidence structure, color graph, strongly regular graph, distance regular graph, spread in a known graph, embedding of one object in another – all these tasks were performed many times during the course of this project. Additional such tasks include computation of an automorphism group, its rank, its orbit structure in a speciﬁed action, isomorphism-testing, and so on. In all cases the obtained results were computer-dependent, that is, the object or its property was elaborated in terms of some routine data generated by machine. Thus, ultimately, we were faced with the important task of performing a posteriori reasoning, in order to get a description of the resulting object or property which was both clear and friendly. We distinguish below two levels of such description. Suppose, for example, we obtain a computer-generated description of an incidence structure S = (P, B). By an explanation of S, we mean a lucid computer-free description of P , B, and the incidence between them. Essential use of a computer, or of additional hand calculations, is not required in this case. By an interpretation of S, we mean that in addition to an explanation we have also a clear self-contained proof that S indeed has the properties it is purported to have (for example, S is a Steiner system with parameters t, k, v). Ideally, an interpretation should be reasonably short, aesthetically pleasing, and methodologically clear. We can also speak about a “conditional interpretation,” in which the proof depends on some well established and reliable source of information (for example, a catalog). In this paper we provide a number of explanations and interpretations. As a rule, we prefer to allow the reader to decide which level of description has been achieved, due to the delicate and subjective nature of the distinction we have set forth.

4 Siamese Objects: Main Deﬁnitions 4.1 Siamese Color Graphs Deﬁnition 12. Let W = (V, {IdV , S, R1 , R2 , . . . , Rn }) be a color graph for which (1) (V, S) is an imprimitive disconnected srg, i.e., a partition of V into cliques of equal size. In what follows, it will be called a spread. (2) For each i, graph (V, Ri ) is an imprimitive drg of diameter 3 with antipodal system S. (3) For each i, (V, Ri ∪ S) is an srg.

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Then W is a Siamese color graph. We call S the spread of Γ , and the number n of drg’s the Siamese rank of W . Given a Siamese color graph W we indicate by (v, k, λ, μ, σ) its parameter set, where (v, k, λ, μ) is the (common) parameter set of each srg (V, Ri ∪S) and σ is the valency of the spread S. There are obvious necessary conditions which must be satisﬁed by such parameters, and we refer to any set (v, k, λ, μ, σ) which satisﬁes these conditions as feasible. Siamese color graphs were ﬁrst studied by Kharaghani and Torabi [35]. The word “Siamese” comes from the observation that any two of the strongly regular graphs share the spread S, so are like conjoined twins. However, after surgical removal of the spread, both “twins” can live an independent life as distance regular graphs. 4.2 Siamese Association Schemes Deﬁnition 13. An association scheme W = (V, {IdV , S1 , . . . , Sn , R1 , . . . , Rk }) is a Siamese association scheme if (V, {IdV , Si , R1 , . . . , Rk }) is a Siamese color graph. In other words, we allow the spread to be a union of basis relations of the scheme. Alternatively, to any association scheme there corresponds a color graph. In this context, the spread in a Siamese association scheme is allowed to be comprised of many colors, provided these colors do not occur anywhere outside the spread. Consequently, given a Siamese color graph one may ask whether it is coming from a Siamese association scheme. We say in this case that the Siamese color graph admits a Siamese association scheme. Deﬁnition 14. A Siamese color graph is said to be geometric if each srg (V, Ri ∪ S) is the point graph of a suitable generalized quadrangle. It is called pseudo-geometric if its parameter set coincides with that of a geometric Siamese color graph (cf. Theorem 2 below). 4.3 Siamese Steiner Designs Proposition 2. Let W be a Siamese color graph with the parameters 4 q −1 , q(q + 1), q − 1, q + 1, q + 1 . q−1 Further assume W is geometric. For each point graph (V, Ri ∪ S) construct a corresponding generalized quadrangle. Let B denote the union of all lines in all resulting GQ’s. Then the incidence structure

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S = (V, B) is a Steiner design

q4 − 1 S = S 2, q + 1, . q−1

Thus a geometric Siamese color graph provides a Steiner system with a spread, and a partition of the remaining blocks into sets which together with the spread form generalized quadrangles. We will call this a Siamese partition of the Steiner system. We further call such a partition coherent if the color graph admits a Siamese association scheme. It is easy to see that a Siamese partition of a Siamese Steiner system provides a geometric Siamese color graph W . 4.4 Pattern of Investigation Given a Siamese color graph W , we can deﬁne many derived combinatorial objects. In order to investigate them, we try to describe their automorphism groups. There are quite a few groups of interest to consider: • the automorphism group Aut(W ), • the color group CAut(W ), • the normalizer group N (W ) (deﬁned to be the normalizer in SV of Aut(W ), where V is the vertex set of W ), • the automorphism groups of the distance regular graphs, • the automorphism groups of the strongly regular graphs. If Γ is geometric, we have in addition • the automorphism groups of the GQ, • the automorphism group of the Steiner system, • the automorphism group of its Siamese partition. Finally, if s = t or s = t + 2 then each srg deﬁnes a symmetric design, which gives us one more group to consider. In fact, not all of these groups are distinct. The automorphism group of a GQ coincides with that of its point graph; if W admits a Siamese partition, then the automorphism group of the partition coincides with CAut(W ); if W admits a Schurian Siamese association scheme, then CAut(W ) coincides with N (W ). In what follows, we restrict ourselves to the case s = t. 4.5 Siamese Graphs as Simultaneous Antipodal Covers Let Γ be a graph. Suppose Γ has an equitable partition Π (e.g., see [9] for a deﬁnition) such that

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• each class (i.e., partition cell) is a coclique, • each point is adjacent to at most one point from each class. In other words, the subgraph induced by any two classes is either empty or a matching (i.e., bipartite and regular of valency 1). We can now deﬁne a graph Δ on Π by joining two classes if there are edges between them. Thus, Δ is the quotient graph Γ/Π. In this case we say that Γ is a cover of Δ. It is called an n-fold cover if each class in Π has n elements. In the particular case in which Γ is an antipodal drg and Π is its antipodal system, we can speak of an antipodal cover of Δ. The distance regular graphs that occur in Siamese color graphs are such antipodal covers, see [9] for their detailed study. A seminal result here is the following theorem due to A. E. Brouwer: Theorem 2. Let Γ be a pseudo-geometric GQ(s, t)-graph with a spread. Then removing the spread from Γ gives a distance regular graph which is an (s + 1)fold cover of the complete graph Kst+1 . Conversely, any drg which is an (s+1)fold cover of the complete graph Kst+1 may be obtained by removing a spread from a suitable pseudo-geometric GQ(s, t)-graph. This gives the following interpretation of Siamese color graphs: The distance regular graphs in a Siamese color graph form simultaneous antipodal covers of Kst+1 which partition the edges of the complete multipartite graph S, where S is the spread.

5 Review of Main Results Part of the results appearing in this paper were for the ﬁrst time brieﬂy reported in [38]. As previously mentioned, an extended abstract of main accomplishments may be found in [39], while a more comprehensive treatment is given in [55]. Currently, we are in the process of preparing papers [40] and [41], which, we hope, will provide a concise, complete, and rigorous introduction to Siamese combinatorial objects, an area which is developmentally in its infancy. Thus, we make no pretense that the current text serve as a complete account of the subject. Recall two of our earlier stated goals for this paper: to involve the reader into the history and logistics of our computer-aided search for Siamese objects, and to share the excitement we experienced from the many observations and discoveries which resulted from this search. Naturally, these goals remain intact. They will serve to motivate the remainder of our text, beginning with the following short summary which incorporates as a whole, the results of our main activity. We started by considering as our initial example the Siamese color graph on 15 points given in [35], which we interpreted group theoretically as a transitive

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action of the group A5 of degree 15. As a second example, we considered a suitable action of A6 on 40 points, and found one more Siamese color graph, which (as was the case for the one on 15 points) admitted a Siamese association scheme. At this stage, we were already able to extrapolate on an evident observation, namely A5 ∼ = PSL(2, 4) and A6 ∼ = PSL(2, 9). Thus, we began an investigation of analogous imprimitive actions of the groups PSL(2, q 2 ) on q 4 −1 q−1 points. Subsequent additional experimentation performed on the cases q = 4, 5, 7 soon convinced us that we were on the threshold of an inﬁnite series of Siamese objects related to the groups PSL(2, q 2 ), the generalized quadrangles W (q), and the classical Steiner designs P G(3, q). Analyzing various sources of information from algebraic combinatorics [7, 47], ﬁnite geometry [30], and group theory [16, 20, 66], we next prepared an outline of a proof of the following: Theorem 3. For each prime power q, there exists an imprimitive action of PSL(2, q 2 ) of degree q 3 + q 2 + q + 1 for which the corresponding association 4 −1 scheme of 2-orbits is a geometric Siamese association scheme on v = qq−1 points. It is easy to show that the resulting Siamese Steiner design is isomorphic to P G(3, q). Also, it is well known that Aut(P G(3, q)) = PΓL(4, q). Due to the inclusion PSL(2, q 2 ) ≤ PΓL(4, q), we next observed that the formerly described action of PSL(2, q 2 ) on v points (the projective points of P G(3, q 2 )) induces another action of PSL(2, q 2 ) on the projective lines of P G(3, q 2 ). Thus we obtained: Corollary 1. Consider the transitive action of PSL(2, q 2 ) on the points of P G(3, q) which results from the inclusion PSL(2, q 2 ) ≤ PΓL(4, q). Next consider the induced action of PSL(2, q 2 ) on the lines of P G(3, q). Then the orbits of this latter action admit a Siamese partition of P G(3, q). This is brieﬂy how we came about an inﬁnite series of Siamese objects (i.e., color graphs, association schemes, Steiner designs, and coherent Siamese partitions). Note that the list of references provided above is not complete; its function is merely to indicate to the reader the origins of some crucial ideas. We refer to this series as “classical,” despite the fact that its presentation has never appeared before [38] and [55]. Our reasoning is that everything needed for the construction of the series, and all subsequent veriﬁcation of its correctness, is readily available as classical results, especially those borrowed from group theory. We also think that the material presented here sheds some new light on classical isomorphisms. Implicitly, a few such isomorphisms will be touched upon in subsequent sections of this paper; however, we make no attempt to treat these isomorphisms in a totally rigorous and comprehensive manner.

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6 Initial Example on 15 Points Recall once more the early part of our story: We worked with the text [35], which presents in evident form the adjacency matrix of a color graph on 15 points. Using GAP, we found the automorphism group of this color graph, which corresponds to the unique (up to similarity) imprimitive representation of A5 of degree 15. From there we constructed the centralizer algebra of this representation, and proved that a certain merging of its classes produced the color graph from [35]. Below we repeat this portion of the job; in particular, we give an explicit description of the centralizer algebra. 6.1 Data from COCO Let us start with the following generators of A5 : A5 = (0, 1, 2, 3, 4), (2, 3, 4). We consider the 1-factor of the complete graph on the vertex set {0, 1, 2, 3, 4} given by x = {{0, 1}, {2, 3}}, with isolated vertex 4. Clearly, Aut(x) = D4 , where D4 = (0, 2, 1, 3), (0, 1) is the dihedral group of order 8. Note that the intersection D4 ∩ A5 coincides with an elementary abelian group E4 =

(0, 1)(2, 3), (0, 2)(1, 3) of order 4. Consequently, if we set Ω = xA5 (where xA5 is the A5 -orbit containing x), then the action of A5 on Ω is similar to its action on the coset space A5 /E4 . That is, (A5 , Ω) is the desired imprimitive action of degree 15. The numeration of elements of Ω given in Table 2 was internally generated by COCO, and we shall adopt it throughout. (For example, “0” is COCO’s designation for the element {{0, 1}, {2, 3}} ∈ Ω.) Starting with the induced generators of the permutation group (A5 , Ω), we get a description of the scheme M = (Ω, 2-orb(A5 , Ω)). Namely M has ﬁve classes R1 , R2 , R3 , R4 , R5 of respective valencies 4, 4, 4, 1, 1. Of these, R4 , R5 form a pair of antisymmetric classes. Another function of COCO provides a list of the intersection numbers of M, i.e., structure constants of the centralizer algebra V (A5 , Ω). Note that analysis of these structure constants shows that V (A5 , Ω) is a non-commutative adjacency algebra, in other words, M is a non-commutative association scheme. Table 2. Elements of Ω as internally generated by COCO 0 {{0,1},{2,3}} 5 {{0,4},{1,2}} 10 {{0,2},{1,3}}

1 {{1,2},{3,4}} 6 {{0,1},{2,4}} 11 {{0,3},{1,2}}

2 {{0,1},{3,4}} 7 {{0,2},{3,4}} 12 {{0,4},{1,3}}

3 {{0,4},{2,3}} 8 {{0,3},{2,4}} 13 {{0,3},{1,4}}

4 {{1,3},{2,4}} 9 {{1,4},{2,3}} 14 {{0,2},{1,4}}

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Finally, we ask COCO to describe all fusion schemes of M and their respective automorphism groups. In particular, we get that Aut(M) = A5 (that is, (A5 , Ω) is a 2-closed permutation group). Regarding proper fusion schemes, we obtain that (Ω, R4 ∪ R5 ) deﬁnes an imprimitive strongly regular graph of the form 5◦K3 , while each of the graphs (Ω, Ri ∪R4 ∪R5 ), 1 ≤ i ≤ 3, provides an srg(15, 6, 1, 3) having automorphism group of order 720. Thus, we are able to prove with the aid of COCO that M is a (Schurian) Siamese association scheme. For the reader’s convenience, we provide an explicit list of the 2-orbits of (A5 , Ω) as follows: R1 = (0, 1)A5 , R2 = (0, 2)A5 , R3 = (0, 4)A5 , R4 = (0, 10)A5 , R5 = (0, 11)A5 . Though all of these results may be obtained a posteriori without the aid of a computer, we do not feel it is prudent to challenge the reader to fulﬁll these computations independently. One of our reasons is that this computerfree task may be accomplished instead by some quite beautiful theoretical considerations. Below we restrict ourselves to only a brief outline of such activities; for details, the reader is referred to [40, 41] and [55]. 6.2 Theoretical Interpretation The existence of the Siamese association scheme M above implies the existence of a Siamese ST S(15) which we denote by S. It is well known that Aut(S) ∼ = PSL(4, 2). In fact, one is able to es= A8 ∼ tablish the exceptional isomorphism A8 ∼ = PSL(4, 2) directly, by simultaneous consideration of the classical geometric model, which gives Aut(P G(3, 2)) = PSL(4, 2), and the sporadic combinatorial model (consisting of an A8 -orbit of aﬃne designs AG(3, 2) and all partitions of an 8-element set into two parts of equal size), which gives Aut(P G(3, 2)) = A8 . We list here some additional striking facts which are required by us (see also 7.1): • Up to isomorphism, there exists a unique generalized quadrangle of order 2, hence it is self-dual and isomorphic to W (2). • The classical model of W (2) is provided by all 2-element subsets of {1, 2, 3, 4, 5, 6} (as points) and all partitions of {1, 2, 3, 4, 5, 6} into three 2-element subsets (as lines), cf. Example 3. • The complement T (6) of the triangular graph T (6) is the unique srg(15, 6, 1, 3). • T (6) is geometric (namely, it is the point graph of W (2)), and Aut(T (6)) = S6 . • Up to isomorphism, there exists a unique antipodal drg of valency 4 and diameter 3 on 15 points, namely the line graph of the Petersen graph. Altogether, this classical information allows us to prove that S is iso morphic to P G(3, 2). Moreover, we can prove that S has 83 = 56 diﬀerent Siamese partitions. Let τ be one such partition. Then Aut(τ ) is isomor-

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phic to the stabilizer of a 3-element subset of {1, . . . , 8} in the natural action (A8 , {1, . . . , 8}); in other words, Aut(τ ) is isomorphic to (S5 × S3 )+ , where we indicate by G+ the subgroup of all even permutations in group G. Although a more detailed account of this result is beyond the scope of this paper, we nevertheless are able to present in Sect. 7 an alternative interpretation of the group (S5 × S3 )+ which is essentially computational in nature, and which ﬁts very naturally within the frames of our current exposition. 6.3 A Few Words About STS (15) A complete listing of all STS (15) was elaborated quite a long time ago in the paper [13], where a catalog of 80 isomorphism classes of such combinatorial designs was given, together with suﬃciently detailed arguments to establish its completeness. Later on, the same problem was attacked by R. A. Fisher [19] who determined only 79 isomorphism classes. At the dawn of the computer era, however, the result of [13] was conﬁrmed in [27]. Nowadays, information about the 80 isomorphism classes of STS (15) is regarded as one of the classical sources in design theory. The paper [48] provides a detailed catalog of these designs together with their numerous properties. We will refer many times to this classical source, in particular we will henceforth adopt their notation STS (15)#1 for the design S = P G(3, 2). An interesting observation was exploited in [36], where the authors considered a certain kind of switching (sometimes referred to a “Pasch switching,” e.g., see [24]) which transforms an initial STS (15) into a second, possibly nonisomorphic, STS (15). If we construct a graph whose vertex set consists of the 80 isomorphism classes of STS (15), with two vertices adjacent if and only if a representative of the ﬁrst class may be switched to a representative of the second class, then this graph will have two connected components: one of size 79 (which includes STS (15)#1 as a vertex of valency one), and one isolated vertex. It is the class associated with this isolated vertex that totally escaped the attention of Fisher in [19]. Following [48], we denote this “isolated” class by STS (15)#80. At this point we would like to make a quite important observation. Among all 80 STS (15), there are exactly two which have point-transitive automorphism group: STS (15)#1 (with automorphism group A8 ), and STS (15)#80 (with automorphism group of order 60). We will discuss other properties of STS (15) as the need arises in the course of our presentation.

7 Automorphism Group of a Siamese Partition for STS (15)#1 Although the reader may be justiﬁed in regarding this section as somewhat tangential to our task, we have included it in order to fulﬁll a broader mission

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of our presentation: to acquaint the reader with the methodology of experimentation, and to explain what is meant by computer algebra experimentation “in action.” We start by delineating the rules of the game. We work with the Siamese association scheme M on 15 points and the corresponding Siamese design S = P G(3, 2). Denote by τ = τ (S) = (P, {B1 , B2 , B3 , S}) a Siamese partition of S. Here P is the point set of S (replacing the earlier notation Ω for the point set of M), S is a spread, and Bi ∪ S is the set of lines of GQ(2)#i, i ∈ {1, 2, 3}. Let N = Aut(τ ) be the automorphism group of the partition τ . We know already from two independent sources (via computer, and via theoretical reasonings) that Aut(τ ) ∼ = (S5 × S3 )+ . We agree to postpone a complete theoretical description to the sequel [41], preferring here to give a self-contained “naive” explanation of N . 7.1 Summary of Known Results We alert the reader that throughout this section the permutation group (A5 , Ω) of Sect. 6.1 will henceforth be denoted by (A5 , P ), commensurate with our choice of notation above. 7.1.1 Association scheme M is clearly obtained as the scheme of 2-orbits of (A5 , P ). As a by-product of our activity, we will ﬁnd an alternative way to establish that Aut(M) = A5 . By construction, A5 acts transitively on P . In what follows, let P3 denote the set of all 3-element subsets of P . We wish to evaluate the number t3 of orbits of (A5 , P3 ). For this goal we use a well known orbit-counting lemma (see [37, 18] for its formulation, and some insightful examples of its combinatorial applications). We denote by Z(H, X) the cycle index polynomial of the permutation group (H, X). 1 (x51 +15x1 x22 +20x21 x3 +24x5 ). From this, Clearly, Z(A5 , {0, 1, 2, 3, 4}) = 60 it is easy to establish that 1 15 x1 + 15x31 x62 + 20x53 + 24x35 . 60 1 Thus, we obtain t3 = 60 ( 15 3 + 15(1 + 18) + 20 · 5) = 14. We conclude that P (A5 , 3 ) has 14 orbits. Z(A5 , P ) =

7.1.2 Clearly, we may identify the elements of P with the edges of the Petersen graph. Moreover, from the indicated representative of R2 (cf. Sect. 6.1), we immediately conclude that the graph Γ2 = (P, R2 ) is none other than the line graph of the Petersen graph. (In fact, one can ﬁnd this same interpretation of Γ2 occurring already on page 1 of [9].) Finally, simply recall that Aut(Γ2 ) ∼ = S5 (e.g., use the famous Whitney-Jung Theorem, see [28]).

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We now wish to repeat our former computation of t3 , only this time replacing A5 by S5 . This initially gives that Z(S5 , {0, 1, 2, 3, 4}) equals 1 5 x + 10x31 x2 + 20x2 x3 + 30x1 x4 + 15x1 x22 + 20x21 x3 + 24x5 120 1 from which it follows that Z(S5 , P ) =

1 15 x1 + 10x31 x62 + 20x3 x26 + 30x1 x2 x34 + · · · . 120

From here we proceed to compute P 15 1 t3 S5 , = + 10(1 + 18) + 20 · 1 + 30 · 1 3 120 3 + 15(1 + 18) + 20 · 5 = 9, in other words, group (S5 ,

P 3

) has 9 orbits.

7.1.3 We know that each srg in our Siamese association scheme M is isomorphic to T (6) and has automorphism group S6 . In this case, P is the vertex set of T (6). Listing of the orbits of (S6 , P3 ) has already been accomplished, see Example 4. 7.1.4 If we now permit ourselves a glance at the catalog [48], we easily identify the Steiner design of Example 4 as STS (15)#1; indeed, it is the only STS (15) with automorphism group of order properly divisible by 60. However, it turns out that such identiﬁcation is not crucial for the fulﬁllment of our remaining steps. 7.1.5 Because M is a Schurian association scheme, it is easy to show that N coincides with the normalizer in S15 of the 2-closure of (A5 , P ). Again, appealing to [48] we see that Aut(S) contains a cycle of length 15, and we are able to show that such a cycle belongs to N . (Important note: In future considerations of our classical inﬁnite series, existence of a similar cycle in each case may be extracted from the famous Singer Theorem, which states that Aut(P G(3, q)) contains a cyclic subgroup which acts transitively on points.) Nevertheless, we can avoid entirely such considerations here by using GAP, which identiﬁes N as a group of order 360 which is isomorphic (as an abstract group) to (S5 × S3 )+ . We shall use this information as our real starting point. Our goal is to better understand the group (N, P ). In what follows, we shall regard S5 and S3 in their natural actions on {0, 1, 2, 3, 4} and {5, 6, 7}, respectively.

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7.2 Other Roads to Group N Before we continue with our exploration of group N , we would like to mention that there are at least two additional ways to approach its identiﬁcation as an abstract group, which do not rely on computer: • Identiﬁcation of Aut(S) as A8 , see [40, 41]. • Combined use of two portions of group theoretic information, namely Z15 ≤ N and S5 ≤ N (the latter coming from our knowledge of Aut(Γ2 )). Together, these provide the ingredients for a computer-free derivation of the structure of N . 7.3 Subdirect Products We need to brieﬂy recall the important group theoretic notion of subdirect product. Let H1 , H2 , and M be groups, and let f1 : H1 → M , f2 : H2 → M be homomorphisms onto M . Consider the set H ⊂ H1 × H2 deﬁned by (a, b) ∈ H if f1 (a) = f2 (b) for a ∈ H1 , b ∈ H2 . Then H is a group, called the subdirect product of H1 and H2 with respect to the homomorphisms f1 , f2 . The order of H is given by |H1 | · |H2 | . |H| = |M | More information on subdirect products may be found in [60] and [56]. Example 6. We illustrate by simple example the notion of subdirect product. Consider H1 = (1, 2) ∼ = S2 and H2 = (3, 4) ∼ = S2 in their natural actions on {1, 2} and {3, 4}, respectively. Thus, the group G = H1 ×H2 acts intransitively on {1, 2, 3, 4}. Let us describe all subgroups of order 2 in G. Evidently, we have the initial subgroups H1 and H2 (up to identiﬁcation with their embeddings), however we have one more group H = (1, 2)(3, 4). In fact, group H is a subdirect product, with respect to the trivial homomorphisms from H1 and H2 onto S2 . A more interesting example will be discussed below. 7.4 Faithful Actions of N on 15 Points We know our group N acts faithfully on 15 points. Here we want to further investigate and interpret this action. Clearly, such an action is similar to (N, N/K), where N/K is the coset space of an appropriate subgroup K ≤ N . Moreover, we know |K| = 24, and further that K is an anti-invariant subgroup of N (meaning that K does not contain any non-trivial normal subgroup of N ). An obvious candidate for K is a group (S4 × S2 )+ where, for example, the evident factors act on {0, 1, 2, 3} and {5, 6}, respectively.

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At ﬁrst glance it would appear that this candidate fulﬁlls all requirements. Indeed, it is not hard to show that the resulting permutation group (N, N/K) is similar to a subgroup of index two in the direct product of permutation groups S5 and S3 (see [37] for a discussion of this notion), and more so, that this subgroup is comprised only of even permutations. However, one can now establish that the degree 15 action of a copy of A5 in the resulting transitive action of N must be intransitive with three orbits of length 5. As this contradicts the initial action (A5 , Ω), we must seek another candidate for the role of K. Recall that S4 has a quite exceptional property among symmetric groups Sn , namely it admits S3 as a homomorphic image. (Perhaps the easiest way to see this is to consider the set X of all 1-factors of the complete graph K4 . Clearly S4 = Aut(K4 ) acts transitively on X, which consists of three members.) Thus, we can construct a subdirect product K of S4 × S3 , with respect to homomorphisms f1 : S4 → S3 and f2 : S3 → S3 , which has order 3| = 24. Moreover, it is easy to see that all permutations in |K| = |S4|S|·|S 3| (K, {0, . . . , 7}) are even, whence K ≤ (S5 × S3 )+ . Upon further examination, we conclude that K is indeed an ideal candidate. 7.5 Explicit Desired Action of N on 15 Points There is a notable distinction between natural and induced actions, particularly when describing group generators via computer. Because N is a subgroup of the symmetric group on {0, . . . , 7}, we may regard the action (N, {0, . . . , 7}) as natural and well understood. This is not true, however, of the induced action (N, N/K). Indeed, to understand this action we need to recruit the aid of a suitable combinatorial structure X, deﬁned on the base set {0, . . . , 7}, for which Aut(X) ∩ N = K. This mirrors a general paradigm for describing induced actions and their generators via machine as implemented in COCO. Thus, we let N ∼ = (S5 × S3 )+ be given explicitly by N = (0, 1, 2), (0, 1, 2, 3, 4), (5, 6, 7), (0, 1)(5, 6), and we deﬁne the structure X = {{0, 1, 5}, {2, 3, 5}, {0, 2, 6}, {1, 3, 6}, {0, 3, 7}, {1, 2, 7}}. The reader is now asked to verify that Aut(X) ∩ N = (0, 1, 2, 3)(5, 7), (0, 1)(2, 3), (0, 1, 2)(5, 7, 6) = K. This means that if we now construct the orbit Ω = XN , then |Ω | = 15, and the permutation group (N, Ω ) contains a subgroup A5 whose action on Ω is similar to the desired action (A5 , Ω). Thus, we may identify the elements of Ω with those of Ω .

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More precisely, if we take A5 = (0, 1, 2), (0, 1, 2, 3, 4) then A5 ∩ Aut(X) =

(0, 1)(2, 3), (0, 2)(1, 3) is a subgroup of order 4, in fact it is the same group E4 which appears in Sect. 6.1. Thus our action (A5 , Ω ) is transitive and similar to (A5 , A5 /E4 ), in other words, it is similar to (A5 , Ω). This implies that after a suitable identiﬁcation of elements, (N, Ω) is an overgroup of (A5 , Ω). Let us brieﬂy explain the genesis of structure X above. Recall from Sect. 6.1 the set X of 1-factors of the complete graph on {0, 1, 2, 3}, namely X = {{{0, 1}, {2, 3}}, {{0, 2}, {1, 3}}, {{0, 3}, {1, 2}}}. We label the members of X by 5, 6, 7 in some speciﬁed order. Subject to this labeling, we now construct X as the set of all {a, b, c} for which {a, b} is an edge in the 1-factor labeled c. In a similar manner, we perform this construction starting with sets of 1-factors arising from other choices of 4-vertex subsets of {0, 1, 2, 3, 4} (just as X arose from the subset {0, 1, 2, 3}). In a sense, group N “coordinates” this job. The objective diﬃculty in the consideration of set Ω is the following: There are 15 · 3! = 90 ways to label elements of Ω (1-factors of K5 with one isolated point) by the elements of {5, 6, 7}. The orbit Ω = XN consists of 1/6 of these labellings. For convenience, we include below an explicit list of the elements of Ω as generated by COCO. We believe that the reader will agree, pending its close examination, that the task of selecting these 15 elements from the entire list of 90 options is not so easy. 0. {{0, 1, 5}, {0, 2, 6}, {0, 3, 7}, {1, 2, 7}, {1, 3, 6}, {2, 3, 5}} 1. {{0, 1, 6}, {0, 2, 7}, {0, 3, 5}, {1, 2, 5}, {1, 3, 7}, {2, 3, 6}} 2. 3. 4. 5. 6.

{{1, 2, 5}, {1, 3, 6}, {1, 4, 7}, {2, 3, 7}, {2, 4, 6}, {3, 4, 5}} {{0, 1, 7}, {0, 2, 5}, {0, 3, 6}, {1, 2, 6}, {1, 3, 5}, {2, 3, 7}} {{1, 2, 6}, {1, 3, 7}, {1, 4, 5}, {2, 3, 5}, {2, 4, 7}, {3, 4, 6}} {{0, 2, 5}, {0, 3, 7}, {0, 4, 6}, {2, 3, 6}, {2, 4, 7}, {3, 4, 5}} {{0, 2, 7}, {0, 3, 6}, {0, 4, 5}, {2, 3, 5}, {2, 4, 6}, {3, 4, 7}}

7. 8. 9. 10. 11.

{{0, 2, 6}, {0, 3, 5}, {0, 4, 7}, {2, 3, 7}, {2, 4, 5}, {3, 4, 6}} {{1, 2, 7}, {1, 3, 5}, {1, 4, 6}, {2, 3, 6}, {2, 4, 5}, {3, 4, 7}} {{0, 1, 5}, {0, 3, 6}, {0, 4, 7}, {1, 3, 7}, {1, 4, 6}, {3, 4, 5}} {{0, 1, 6}, {0, 3, 7}, {0, 4, 5}, {1, 3, 5}, {1, 4, 7}, {3, 4, 6}} {{0, 1, 7}, {0, 3, 5}, {0, 4, 6}, {1, 3, 6}, {1, 4, 5}, {3, 4, 7}}

12. {{0, 1, 7}, {0, 2, 6}, {0, 4, 5}, {1, 2, 5}, {1, 4, 6}, {2, 4, 7}} 13. {{0, 1, 5}, {0, 2, 7}, {0, 4, 6}, {1, 2, 6}, {1, 4, 7}, {2, 4, 5}} 14. {{0, 1, 6}, {0, 2, 5}, {0, 4, 7}, {1, 2, 7}, {1, 4, 5}, {2, 4, 6}} 7.6 Analytic Enumeration of Orbits of (N,

Ω 3

)

We now count the orbits of the action of N on 3-element subsets of Ω. Because this task requires more comprehensive eﬀorts, we organize our computations

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Table 3. Cycle index data for group (N, Ω) # 1 2

g1 ∈ S5 e

g2 ∈ S3 e

g∈N e

e

(5, 6, 7)

(5, 6, 7)

|ccl(g)| 1 2

CI1 x81

CI2 x15 1

x51 x3

x53 x31 x62

19

x31 x62

19

x3 x26

1

x3 x26 x53 x53

1

x1 x2 x34

1

6

(0, 1, 2)(3, 4)

(5, 6)

(0, 1, 2)(3, 4)(5, 6)

60

x41 x22 x41 x22 x1 x22 x3 x1 x22 x3

7

(0, 1, 2)

e

(0, 1, 2)

20

x51 x3

3

(0, 1)

(5, 6)

(0, 1)(5, 6)

30

4

(0, 1)(2, 3)

e

(0, 1)(2, 3)

15

5

(0, 1)(2, 3)

(5, 6, 7)

(0, 1)(2, 3)(5, 6, 7)

30

χ(g) 15 3

5

5

8

(0, 1, 2)

(5, 6, 7)

(0, 1, 2)(5, 6, 7)

40

9

(0, 1, 2, 3)

(5, 6)

(0, 1, 2, 3)(5, 6)

90

x21 x23 x21 x2 x4

x35

0

x15

0

10

(0, 1, 2, 3, 4)

e

(0, 1, 2, 3, 4)

24

x31 x5

11

(0, 1, 2, 3, 4)

(5, 6, 7)

(0, 1, 2, 3, 4)(5, 6, 7)

48

x 3 x5

5

with the aid of Table 3. (In it, e denotes the identity group element, |ccl(g)| the size of the N -conjugacy class containing g, CI1 the contribution of g to the cycle index polynomial Z(N, {0, . . . , 7}), CI2 the contribution of g to the cycle index polynomialZ(N, Ω), and χ the permutation character corresponding to the action (N, Ω3 ).) From the data in Table 3, we now get that t3 (N, Ω) is equal to 1 (5 · 7 · 13 + 2 · 5 + (45) · 19 + (90) · 1 + (60) · 5 + 90 · 1) = 5. 360 Thus, using only the orbit-counting lemma we are able to determine that (N, Ω3 ) has precisely 5 orbits. 7.7 Constructive Enumeration of Orbits of (N,

Ω 3

)

We would like now to physically construct the orbits enumerated in Sect. 7.6, by which we mean exhibit at least one representative for each orbit, determine each orbit’s length and, ideally speaking, explain its structure. The results of such enumeration are presented below, in terms of the set Ω = XN (cf. Sect. 7.5). To eliminate mistakes, this activity was performed with the aid of a computer; however, in principle one can avoid such usage during the explanation stage of enumeration. From this information we see that the only available option to form 35 blocks of S is to take the union of orbits 1 and 5 (see Table 4). Although it is not requested for our goals, perhaps the easiest way to explain these orbits is to perform a constructive enumeration of the 9 orbits of the group S5 (see Sect. 7.1.2), and then merge these to form the orbits of N . One explanation for this is that the enumeration of the 9 orbits of S5 is a rather simple task,

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Orbit# 1 2 3 4 5

Length 30 180 180 60 5

Representative {0, 9, 13} {0, 13, 14} {0, 10, 13} {0, 5, 13} {0, 1, 3}

due to the fact that these orbits are in bijective correspondence with the isomorphism classes of 3-edge subgraphs of the Petersen graph. The following facts may also be useful in this endeavor: • S5 is normal in N with quotient group Z3 . Thus, there are only two possibilities for a given orbit of N : either it is an orbit of S5 , or it is fused from three S5 -orbits of equal length. • A cleverly arranged bijection between Ω and Ω could greatly simplify the job. We stress that in our eyes this last activity has very high methodological signiﬁcance, in fact much more so than the practical signiﬁcance of the resulting explanation (or even interpretation) of the analytically enumerated orbits of (N, Ω3 ). 7.8 Summary of Results About N For the reader’s convenience, we have compiled below all objective results from Sects. 6 and 7 about the group (N, Ω) and its related structures. We will not burden the reader with a further discussion of our reasonings, other than to say that a rigorous proof depends strongly on the “rules of the game,” as set forth in the preamble to Sect. 7. Finally, we again mention that an alternate proof (more literate for a group theorist), based on the exceptional isomorphism between A8 and PSL(4, 2), will be presented in [40] and [41]. Proposition 3. M is a Schurian Siamese association scheme. S is a coherent Siamese STS (15). Aut(S) = A8 . N = NS15 (A5 ) ∼ = (S5 × S3 )+ . N is the automorphism group of a Siamese partition of S. Aut(M) = A5 . N acts transitively on the point set of S, and has two orbits on the block set of S of respective lengths 30 and 5. (h) N coincides with the stabilizer in Aut(S) of a spread in S.

(a) (b) (c) (d) (e) (f ) (g)

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Table 5. Representatives of 2-orbits of A4 Ψ1 (e, e) Ψ7 (e, (0, 1)(2, 3)) Ψ12 (e, (0, 2)(1, 3)) Ψ17 (H, (1, 3, 2))

Ψ2 (e, (1, 3, 2)) Ψ8 (e, (0, 2, 1)) Ψ13 (e, H) Ψ18 (H, (1, 2, 3))

Ψ3 (e, (1, 2, 3)) Ψ9 (e, (0, 1, 2)) Ψ14 (e, H(1, 2, 3)) Ψ19 (H, H)

Ψ4 (e, (0, 3)(1, 2)) Ψ10 (e, (0, 3, 2)) Ψ15 (e, H(1, 3, 2)) Ψ20 (H, H(1, 2, 3))

Ψ5 Ψ6 (e, (0, 2, 3)) (e, (0, 1, 3)) Ψ11 (e, (0, 3, 1)) Ψ16 (H, e) Ψ21 (H, H(1, 3, 2))

8 More About 15 Points To illustrate the fact that the notion of a Siamese scheme is stronger than that of Siamese color graph, we give an example of a Siamese color graph which does not admit a Siamese association scheme. 8.1 Starting Group We consider the group A4 , a one-point stabilizer in A5 , in its intransitive action on the set Ω as deﬁned in Sect. 6.1. The group (A4 , Ω) has two orbits V1 , V2 of respective lengths 3 and 12. Evidently, A4 acts regularly on V2 ; hence it is convenient to identify the elements of V2 with those of A4 . To make our representation consistent, we also describe the elements of V1 in group theoretic terms, namely as cosets of the unique Sylow 2-subgroup H in A4 . Very simple reasoning shows that the action of A4 on V1 ∪ V2 has rank 21. We choose representatives of the 2-orbits of (A4 , V1 ∪ V2 ) as shown in Table 5 (where e denotes the identity element of A4 ). Now we deﬁne the following binary relations on V = V1 ∪ V2 : IdV S Φ1 Φ2

= = = =

Ψ1 ∪ Ψ19 Ψ8 ∪ Ψ9 ∪ Ψ20 ∪ Ψ21 Ψ5 ∪ Ψ10 ∪ Ψ12 ∪ Ψ14 ∪ Ψ18 Ψ6 ∪ Ψ7 ∪ Ψ11 ∪ Ψ15 ∪ Ψ17

Φ3 = Ψ2 ∪ Ψ3 ∪ Ψ4 ∪ Ψ13 ∪ Ψ16 Finally, we consider the color graph W = (V, {IdV , S, Φ1 , Φ2 , Φ3 }). Proposition 4. (a) Aut(W ) = A4 , (b) W is a geometric Siamese color graph of Siamese rank 3. However, W does not admit a Siamese association scheme.

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8.2 A Non-coherent Siamese Partition of STS (15)#7 Because W is a geometric Siamese color graph, its existence implies that of a Siamese STS (15). We constructed this Siamese Steiner system, and realized that it is isomorphic to STS (15)#7, in the notation of [48]. According to [48] STS (15)#7 has group of order 288 with orbits of length 3 and 12 as its points. Using GAP, we identiﬁed this group as (S4 × S4 )+ in its natural action on 8 points. This allowed us to obtain an interesting model of STS (15)#7 in which there are two types of points and three types of blocks. To better describe this model we start from the action of S4 × S4 on O1 ∪ O2 , where the two evident copies of S4 act naturally and independently on O1 = {1, 2, 3, 4} and O2 = {5, 6, 7, 8}. 8.2.1 Description of the Model of STS (15)#7 Points of the ﬁrst type will be partitions of O2 of the form 2 + 2. There are three such points, and they form a single orbit under the action of (S4 × S4 )+ . To describe points of the second type, we consider all partitions of O1 ∪ O2 of the form 2 + 2 + 2 + 2, where each pair in the partition contains exactly one letter from each of O1 and O2 . There are 4! = 24 such partitions; however we require only half of these. As these 24 partitions fall into two orbits under the action of (S4 × S4 )+ , we deﬁne points of the second type to be the partitions in one of these orbits, say the one which contains the partition {{1, 5}, {2, 6}, {3, 7}, {4, 8}}. Thus in total we have 3 + 12 = 15 points. Blocks of the ﬁrst type will consist of partitions of O1 ∪ O2 of the form 4 + 4 such that each 4-tuple of the partition contains two elements from each 2 of O1 and O2 . The number of such blocks is 12 42 = 18. Blocks of the second type consist of pairs of elements, one element from O1 the other from O2 . There are 42 = 16 such blocks. Finally, we assign one additional block b∞ . Altogether, we get 18+16+1 = 35 blocks. Incidence is deﬁned as follows: A block of the ﬁrst type is incident to exactly one point of the ﬁrst type (the partition it induces on O2 ) and to exactly two points of the second type (the partitions of type 2 + 2 + 2 + 2 which are reﬁnements of the partition of type 4 + 4). A block of the second type is incident to exactly three points of the second type (the partitions which contain it), while block b∞ is incident to all three points of ﬁrst type. Proposition 5. The incidence structure deﬁned above is an STS (15). Moreover, it is isomorphic to STS (15)#7. The second statement of the proposition may be proved by computer, however it also follows easily from theoretical considerations. Indeed, it is evident by construction that our model is invariant with respect to (S4 × S4 )+ , and consulting [48] one sees that STS (15)#1 and STS (15)#7 are the only STS (15) which admit a group of automorphisms of that order. By using certain invariants provided in [48], the task is easily completed.

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Now we wish to describe a Siamese partition of STS (15)#7. For this purpose, let {x, y, z} be an arbitrary 3-element subset of O1 . Starting from the pair {x, y}, let B1 consist of all blocks of the ﬁrst type in which these two elements appear in the same partition cell. Let B2 consist of all blocks of the second type which contain either x or y. Finally, let Ω denote the set of all points of the Steiner system and set B = B1 ∪ B2 ∪ {b∞ }. Of course, in a similar manner we could deﬁne a corresponding incidence structure starting from {x, z} or {y, z}. Note also that there are 43 = 4 choices for the initial selection of a 3-element subset from O1 . Proposition 6. (a) Each of the incidence structures deﬁned with respect to {x, y}, {x, z} and {y, z} is a GQ(2). (b) Any two of these three generalized quadrangles intersect in the same spread. (c) The generalized quadrangles deﬁned by {x, y, z} form a Siamese partition of STS (15)#7. (d) This Siamese partition is non-coherent (i.e., it is implied solely by the existence of the Siamese color graph W). (e) The automorphism group of the Siamese partition has order 72. 8.3 All Siamese Color Graphs on 15 Points are Obtained Above we gave the construction of two geometric Siamese color graphs on 15 points. To prove that there are no others, the following steps have been taken: • A geometric color Siamese graph on 15 points provides a Siamese partition of an STS (15). • There are 80 non-isomorphic STS (15). • For each STS (15), a computer search was performed to enumerate all embedded GQ(2). • It was checked if three of these generalized quadrangles form a Siamese partition. Thus, we get the following result: Proposition 7 (Computer search). The only STS (15) admitting a Siamese partition are the designs STS (15)#1 and STS (15)#7. In both cases, the partition is unique up to isomorphism. Corollary 2. There are exactly two non-isomorphic geometric Siamese color graphs on 15 vertices. It is well known that the triangular graph T (n) is uniquely determined by its parameters, except for the case n = 8. In particular, T (6) is the only srg(15, 6, 1, 3), and it is the point graph of the unique GQ(2). Thus we get the following:

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Corollary 3. Every Siamese color graph on 15 vertices is necessarily geometric. Hence, there are exactly two non-isomorphic Siamese color graphs on 15 vertices, one of which admits a Siamese association scheme. Remark 2. Using a computer, one could pursue an alternative line of proof which involves constructing all color graphs on 15 points which are “pretending” to be Siamese, then checking these one-by-one for conﬁrmation. We believe, however, that the approach we have outlined above is preferable for the following reasons: • It is more consistent with the goals of our presentation. • It is well controlled because the catalog [48] is a very reliable source of information. • As a by-product, we get a nice opportunity to compare “Siamese properties” with other features of STS (15) (see also the discussion in Sect. 10). • In principle, ad hoc techniques may be used to eliminate most STS (15) from consideration. As a sequel to the above remark, we believe that ﬁnding a computer-free proof of Corollary 3 constitutes a worthwhile and challenging open problem. In this initiative, one could perhaps exploit methods from topological graph theory (in the sense of [26]) to elaborate a proof. In any case, we feel that some rather innovative ideas may be required to remove this computer dependence.

9 Objects on 40 Points 9.1 Classical Objects According to Sect. 5, we are aware of the existence of an inﬁnite series of Siamese objects which we choose to call “classical.” Our interest here is to scrutinize the one on 40 points. As usual, we refer to [40, 41] and [55] for extra details. Recall the following well known facts about generalized quadrangles of order 3 (e.g., see [54, 52, 9, 65, 18]): • Up to isomorphism, there exist only two GQ of order 3: W (3), and its dual Q(4, 3). • W (3) has a unique spread (up to isomorphism), however Q(4, 3) has no spreads. • Aut(W (3)) ∼ = PΓU (4, 2) ∼ = Sp(4, 3).Z2 . • A spread in W (3) is invariant with respect to S6 , which is a maximal subgroup of PSU (4, 2). With the aid of COCO, we constructed a Siamese association scheme on 40 points from the action of A6 ∼ = PSL(2, 9) on the cosets of a ﬁxed Sylow 3-subgroup. This action is not 2-closed; COCO returned its 2-closure as a

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group of order 720 which we subsequently identiﬁed as A6 × S2 . According to the information above, we knew in advance (at both the theoretical and computational levels) that the automorphism group of the unique antipodal geometric drg on 40 points has order 1440. Using a computer, we identiﬁed this latter group as S6 × S2 and gave an a posteriori interpretation of it which we believe to be quite beautiful. Let Ω0 = {1, 2, 3, 4, 5, 6}. Let P be the set of directed cycles of length 3 in Ω0 . Let B1 be the set of all partitions of Ω0 into two triples. Let B2 be the set of directed arcs from Ω0 . Let B = B1 ∪ B2 . Deﬁne incidence as follows: Let a cycle be incident to a partition if its vertex set is one of the partition cells; let it be incident to an arc if it contains this arc. Then we have the following: Proposition 8. (a) The incidence structure (P, B) is a GQ(3). (b) The 10 blocks in B1 provide a spread in this GQ(3). (c) The pair (GQ(3), B1 ) is invariant with respect to S6 × S2 . Remark 3. One can identify a natural involutory automorphism of (P, B) which interchanges two oppositely directed 3-cycles, yet it is not induced from any permutation in S6 . In fact, one can deﬁne the direct factor S2 as being generated by exactly this involution. Corollary 4. The automorphism group of the antipodal geometric drg on 40 points is isomorphic to S6 × S2 . Remark 4. We call attention to a beautiful model of W (3) constructed by S. E. Payne in [53]. Though his model and ours are done in much the same spirit, we believe that ours has a certain advantage, in that it conveys in a very transparent manner the full symmetry of the considered object. We hope that such constructions, and their subsequent analyses, will shed light on higher order Siamese objects, and perhaps lead to a characterization of the classical ones. 9.2 Circulant Example In this section and the next, we introduce two other interesting examples of Siamese color graphs on 40 points, neither of which admits a Siamese association scheme. The one given here may be described in a very transparent manner, since it admits a point-transitive regular cyclic group of automorphisms. Thus, we identify the points with the set Z40 of integers modulo 40. We take the following blocks as basic: b1 = {0, 10, 20, 30} b2 = {0, 1, 6, 32} b3 = {0, 4, 11, 23} b4 = {0, 3, 16, 18}

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and we form the block system B = Bi , where Bi is the orbit of Z40 which contains bi , i = 1, 2, 3, 4. Then we get: Proposition 9. (a) The incidence structure (Z40 , B) is a Steiner system S(2, 4, 40) with a spread. (b) (Z40 , B) is invariant with respect to a subgroup K of order 160 in the full aﬃne group over Z40 , namely K = {x → ax + b | a ∈ {1, 3, 9, 27}, b ∈ Z40 }. (c) Group K is the full automorphism group of (Z40 , B). Note that the reader can easily check parts (a) and (b) of Proposition 9 by hand. We used a computer to verify part (c), though a computer-free proof can also be elaborated. Next, we describe just one of the generalized quadrangles of this system. Consider the subgroup H = {x → ax + b | a ∈ {1, 9}, 4|b}. Take the spread B1 together with the orbits (b2 − 1)H , (b3 + 2)H , and bH 4 under the action of H. Altogether we get 40 blocks which form a generalized quadrangle. The other three GQ are constructed by shifting the given one with the aid of the elements of Z40 . Thus we get: Proposition 10. (a) (Z40 , B) is a (non-coherent) Siamese Steiner design. (b) The stabilizer of a Siamese partition coincides with the group K of Proposition 9. 9.3 One More Point-Transitive Example Consider the group H = A5 × Z4 . We assume that A5 acts naturally on Ω1 = {1, 2, 3, 4, 5}, and that Z4 is generated by the cycle (6, 7, 8, 9) acting on Ω2 = {6, 7, 8, 9}. Thus the base set is Ω0 = Ω1 ∪ Ω2 = {1, 2, . . . , 9}. We now describe points and blocks of the design. In what follows, we shall denote by α the antipode of α ∈ Ω2 with respect to the cycle (6, 7, 8, 9); thus 6 = 8 and 7 = 9. Otherwise, a, b, c, d, e will always denote pairwise distinct elements of Ω1 , while α, β will denote distinct non-antipodal elements of Ω2 . Finally, we abbreviate by aα the pair {a, α}. Points of the design will be pairs {aα, bα }. As there are 52 = 10 choices for a, b ∈ Ω1 and 4 choices for α ∈ Ω2 , there are altogether 40 points, which in fact comprise a single orbit of H. A typical point is {16, 28}. We introduce four types of blocks in our design by way of indicating orbit representatives (see Table 6). The arguments we used to deduce their lengths will be omitted due to space limitations. In all, we have 40 points and 130 blocks in our design. In fact, by way of computer we get the following result:

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Table 6. Block types in terms of orbit representatives Type I II III IV

Length 60 40 10 20

Representative {{16, 28}, {27, 59}, {56, 48}, {47, 19}} {{16, 28}, {18, 46}, {26, 48}, {37, 59}} {{16, 28}, {18, 26}, {17, 29}, {19, 27}} {{16, 28}, {28, 36}, {28, 56}, {28, 46}}

Proposition 11. The structure deﬁned above is a S(2, 4, 40) containing exactly 4 generalized quadrangles which form a Siamese partition. Its automorphism group is generated by A5 × Z4 and the additional automorphism (1, 2)(6, 8); hence it is a group of index two in S5 × D4 . 9.4 Other Siamese Objects The reader is now acquainted with three Siamese designs on 40 points. One is the classical P G(3, 3), whose automorphism group A8 is both point- and block-transitive; the other two designs have groups that act transitively on points, but intransitively on blocks. In fact, 475 such designs were found by us with the aid of a computer, many of which have small automorphism group. Here we are interested in nine such designs which have relatively large group. Table 7 contains information about the groups related to these nine Steiner systems. In it we denote by G the automorphism group of the Steiner system, and by H the automorphism group of the partition (hence, the color group of the corresponding graph). For each group, we also include the lengths of orbits in its actions on points and blocks. Recall that there are exactly two geometric srg(40, 12, 2, 4), one of which is the point graph of W (3). Only this latter srg has a spread, and it is unique up to isomorphism. Therefore, in a geometric Siamese color graph on 40 points there is only one option for the drg and the srg. Note that all Siamese color graphs described here are geometric. Table 7. Some Siamese Steiner designs on 40 points # 1 2 3 4 5 6 7 8 9

S(2, 4, 40) |G| 24261120 160 480 1296 648 288 10368 1296 144

P 40 40 40 4, 36 4, 36 4, 36 4, 36 4, 36 4, 36

B 130 10, 40, 80 10, 20, 40, 60 1, 242 , 81 1, 124 , 81 1, 9, 242 , 362 1, 48, 81 1, 242 , 81 1, 9, 242 , 362

Partition |H| 11520 160 480 144 72 288 576 144 144

P 40 40 40 4, 36 4, 36 4, 36 4, 36 4, 36 4, 36

B 10, 120 10, 40, 80 10, 20, 40, 60 1, 9, 242 , 362 1, 9, 124 , 362 1, 9, 242 , 362 1, 9, 48, 72 1, 9, 242 , 362 1, 9, 242 , 362

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At present, the designs in Table 7 (apart from the ﬁrst three) have only strict computer dependent descriptions, hence we see no sense in disturbing the reader any further with their details. We do however mention a few reasons why we decided to include Table 7 in our presentation: • It can be used in the future for the veriﬁcation of further results on Siamese objects on 40 points, particularly serving as a baseline for the measurement of progress in the area. • Some numerical data appearing in it may be a source of important observations, even conjectures, which may lead later to theoretical interpretations and explanations. • A few objects in the table will be discussed below and again in Sect. 10. Note that the interested reader will ﬁnd a catalog of Siamese partitions for these nine designs in Appendix A.1 of [55]. 9.5 Discussion on Methodology At this point we are well positioned to discuss the methodology used by us for the computer-aided enumeration of all Siamese color graphs of a prescribed order. We mention that while we achieved a complete list of all desired objects on 15 points, we were only partially successful in the case of 40 points. 9.5.1 Strategy A: Combinatorial Analogue of Transitive Extension This was the ﬁrst heuristic approach used by us to construct Siamese objects on 15 and 40 points. Recall that all classical objects described to this point were obtained in a uniﬁed manner: • Start from the action of PSL(2, q 2 ) acting on the q 2 + 1 points of the projective line. 4 −1 . • Construct the corresponding induced transitive action of degree qq−1 • Describe all Siamese color graphs which are invariant with respect to this action. Let us now consider a generalization of this procedure. Namely, at the ﬁrst step we replace PSL(2, q 2 ) by its stabilizer of a projective point (point at inﬁnity). This stabilizer is a certain aﬃne subgroup of PSL(2, q 2 ). With this subgroup we now fulﬁll the second and third steps as described above. Clearly, this generalization will produce all results attainable from the original three steps, however it will hopefully generate other graphs as well, possibly with intransitive automorphism groups. In fact this procedure gave us all Siamese objects on 15 points (here “all” turned out to be just two!), while on 40 points we were able to get a number of such objects (each with automorphism group of order a multiple of 36).

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9.5.2 Strategy B: Construction and Investigation of Steiner Designs The idea behind this second approach is quite simple: Construct as many Steiner designs as possible, and for each isomorphism class of such designs describe all Siamese partitions. As previously mentioned, it was exactly this strategy that enabled us to complete the classiﬁcation of Siamese color graphs on 15 points. In fact, for all but two classes we were able to prove non-existence of Siamese partitions, while for designs coming from each of the two remaining classes (known to be Siamese by Strategy A) we conﬁrmed that a Siamese partition was unique, up to isomorphism. For a given Steiner design, a search for Siamese partitions is achieved in four main steps: • Enumerate all embedded GQ in the Steiner design. • Find all spreads occurring as intersections of the line sets of pairs of such GQ. • For a given such spread, ﬁnd all GQ which contain it. • Among the GQ which contain this spread, determine if the line sets of certain of them (together with spread) form a Siamese partition. The problem of enumerating all Steiner designs on 40 points seems hopeless; thus only special classes of designs were constructed and investigated. The technique used for their construction was the Kramer-Mesner method (see Sect. 2.4). We enumerated all cyclic Steiner systems, as well as those invariant with respect to an intransitive group of order 36, with point orbits of respective lengths 4 and 36. Altogether, using Strategies A and B we were able to construct nine Siamese partitions on 40 points, speciﬁcally the ones which are presented in Table 7. 9.5.3 Strategy C: Direct Enumeration of Siamese Color Graphs This third strategy grew out of our attempts to enumerate all Siamese color graphs on 40 points, when we recognized the need to arrange our search objectives in a more sophisticated way. Currently, the algorithm is formulated strictly for geometric Siamese color graphs, however its extension to the general case should not be extremely diﬃcult to achieve. We provide the following outline. Let Δ be the point graph of W (3), S a ﬁxed spread in Δ, and Γ = Δ \ S the drg obtained by removing the spread S from graph Δ. We set G = Aut(S) and H = Aut(Γ ). In our case, G ∼ = S10 S4 is a huge group of order (4!)10 · 10!, ∼ while H = S6 × S2 is a relatively small subgroup of G. Now each geometric srg Δ intersecting Δ in S may be described via an embedding of an isomorphic copy Γ of Γ inside the complement graph Δ of Δ. Therefore, if we are interested in enumerating all non-isomorphic Siamese pairs (Δ, Δ ) (recall that these may be interpreted as triples (Γ, S, Γ )), then

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we need to enumerate all inequivalent embeddings of Γ into Δ which, in a sense, preserve S. Such enumeration of Siamese pairs forms the ﬁrst stage of our algorithm. Note that this task may be formulated entirely in group theoretic terms: Enumerate all double cosets in G of the form HgH. Indeed, in accordance with our general methodology for implementing COCO, together with the interpretation suggested in Sect. 2.5, we proceed as follows: • Consider the action of G on Γ , where Γ is regarded as a combinatorial structure. • Formulate an action of G on an appropriate coset space G/H which is similar to the action of G on the orbit Γ G containing Γ . • Enumerate all 2-orbits (Γ, Γ )G in terms of the action (G, G/H) (equivalently, enumerate all double cosets HgH in G). At the next stage of the algorithm, we need to extend each obtained pair (Γ, Γ ) to a complete Siamese color graph. Let g ∈ G be a permutation for which Γ g = Γ is embedded into Δ. Then B = Γ ∪ Γ is a regular graph of valency 18. We now remove Δ from the complement graph B of B, giving another regular graph B of valency 18. Now comes a crucial observation in the concrete case we are considering: B admits a partition into two drg with common spread S if and only if B is isomorphic to one of the graphs B (which are presumed to be known after successful fulﬁllment of the ﬁrst stage of the algorithm). Thus each isomorphism between B and B yields a Siamese color graph, and every such graph arises in exactly this manner. Unfortunately, in principle COCO is not able to enumerate all 2-orbits of the action (G, G/H). Equally fortunate, however, is the fact that we do not fully require this; indeed, we are only interested in those 2-orbits (Γ, Γ )G for which the graphs Γ and Γ are disjoint. In any case, both stages of the algorithm were implemented with GAP. The speciﬁc feature of this implementation is that it allowed simultaneous manipulation of both group theoretic and combinatorial information. We will discuss eﬃciency of this algorithm in the next section. At present, we wish only to mention that in order to describe all Siamese color graphs on 40 points we had to initially take into account all antipodal drg having identical spread (not just the geometric ones). In group theoretic terms, this corresponds to replacing the initial transitive action (G, G/H) by a more sophisticated intransitive one. From our experience, implementation and efﬁcient execution of a suitably modiﬁed algorithm would seem to be a quite diﬃcult and time-consuming task. Remark 5. In actuality, our implementation of Strategy C was fulﬁlled at the level of Siamese partitions rather than color graphs. Namely, in place of Δ and other corresponding objects, we considered instead the (unique) GQ(3) having a spread. Taking into account that Aut(GQ(3)) = Aut(Δ), there are no

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changes at all in the group theoretical manipulations involved in the described process, only that the combinatorial criteria used in the selection of suitable double cosets are formulated in slightly diﬀerent terms. In fact, Strategy C as we described it above has deﬁnite advantages over the strategy implemented by us. For one thing, it allows a more plausible generalization to all Siamese color graphs (as brieﬂy discussed in the previous paragraph). 9.5.4 Review of Results on 40 Points As we previously mentioned, a combination of Strategies A and B enabled us to construct nine Siamese color graphs, thus implying the nine Siamese Steiner systems which appear in Table 7 of Sect. 9.4. Further attempts to extend our scope within the conﬁnes of these strategies did not lead to any new Siamese objects. This explains our motivation for creating Strategy C, which in principle turns out to be much more productive than the other ones. Although this algorithm is still in development, much recent progress has been made. For example, running two successive versions of the algorithm, each over a twoday period, gave roughly 130 and exactly 475 Siamese designs, respectively. We ﬁnd this increase very encouraging. Interestingly, for each Siamese Steiner design we investigated it turns out that there is a unique Siamese partition, up to isomorphism. Further note that in [55], in addition to the geometric drg on 40 points with automorphism group S6 × S2 (see Corollary 1), two more drg’s on 40 points were discovered which, in fact, give the smallest possible answer to a question posed by Godsil–Hensel (see [22, 10] for more details). Finally, it was proved in [55] that these two non-geometric drg are the only ones possible. The proof is computer-dependent and is based on inspection of Ted Spence’s catalog of strongly regular graphs (see [63], and also the recent paper [15]). Currently we do not have any example of a Siamese color graph involving either of these two drg; in other words, we do not know an example of a non-geometric Siamese association scheme. 9.5.5 Further Perspectives In principle, it should be possible to enumerate all Siamese color graphs on 40 points, or at least all geometrical ones. Since we found Siamese color graphs related to W (2) and W (3), it is natural to consider W (4) on 85 points. Several heuristic strategies were used to search for Siamese color graphs related to W (4) but only one example was found; it is an association scheme which belongs to our inﬁnite series of classical Siamese objects. While it seems a worthwhile goal to arrange a systematic search for all geometric Siamese color graphs on 85 points, no eﬃcient algorithm for this task has as yet been created. A reasonable guess is that there exist Siamese color graphs on 85 points which do not admit Siamese association schemes.

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However, at present no evidence has surfaced to either support or refute this contention.

10 Additional Remarks There are a number of topics which we regard as essentially important yet only implicitly related to the main line of our presentation. Not wishing to distract the reader’s attention during the initial stages of our message, we instead decided to postpone their discussion to here. 10.1 Theory and Algorithms 10.1.1 Kramer–Mesner Method Though the Kramer–Mesner method was originally presented in [43], it deﬁnitely has numerous predecessors. In particular, the methodology of socalled “tactical decompositions” can be traced to the end of the 19th century, especially to the classical paper [51]. Although such historical roots as these certainly deserve a detailed treatment, this task lies well beyond the scope of the present paper. 10.1.2 Computer Package Discreta In the implementation of the Kramer–Mesner method there are three essential steps: (a) Generate all orbits on t-sets and k-sets. (b) Derive the corresponding Kramer–Mesner matrix. (c) Find all solutions to the related system of Diophantine equations. Since each of these steps requires extensive computations, the Kramer–Mesner method is greatly facilitated by machine. The computer package Discreta was conceived in Bayreuth (Germany) for this main purpose, and to this day it remains, in our judgment, the most powerful software tool for implementation of the Kramer–Mesner method. The project of the development of Discreta was initiated by R. Laue, and a ﬁrst implementation elaborated by B. Schmalz [57]. At the next stage, crucial input was provided by A. Betten and A. Wasserman [6]. The reader is referred to [3, 5] for some beautiful examples of applications of Discreta. We mention that the crucial role of Discreta in the fulﬁllment of step (c) assumes the form of the LLL-algorithm, see [67] for details. 10.1.3 Double Coset Enumeration Another reason for the great success of Discreta is the existence of a special algorithm, called Leiterspiel, which operates at the level of subgroup ladders and double coset graphs. This algorithm goes back to [45], see [58] for an outline of its implementation.

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10.1.4 Automorphism Group of a Design Imagine a design S = (P, B), obtained with the aid of the Kramer– Mesner method via a suitable merging of orbits of a permutation group (H, P ). Clearly, H ≤ G, where G = Aut(S). The task of describing (G, P ) is greatly facilitated by an a priori knowledge of the lattice of overgroups of (H, P ) within the symmetric group deﬁned over set P . This idea was touched upon a few times in the present paper, though in a naive and rudimentary way. In fact, in rigorous form it constitutes a quite deep and interesting topic which lies on the edge between group theory and combinatorics. We refer to [4] for an elementary discussion, and to [34] for a more advanced treatment. 10.1.5 Transitive Extension A familiar concept from group theory, a transitive extension of a permutation group (H, Ω) is a transitive permutation group of the form (G, Ω ∪ {x}), where x ∈ Ω and Gx = H. We also refer to transitive extension when describing the general procedure of determining all groups (G, Ω ∪ {x}) given (H, Ω) as above. See Sect. 3.7.6 of [18] for a brief discussion of this procedure. The notion of transitive extension has a very nice combinatorial generalization. If, in particular, we make the restriction that the groups (G, Ω ∪ {x}) be 2-closed, then we are eﬀectively investigating all regular color graphs with vertex set Ω ∪ {x} which are invariant with respect to (H, Ω). Our Strategy A of Sect. 9.5.1.1 is nothing else but a special case of this “combinatorial” extension. See [33] for what is perhaps the earliest attempt to implement computers along this line of investigation. 10.1.6 Factorization of Graphs Factorization of the complete graph Kv into a number of speciﬁed (regular) v-vertex graphs is a quite general form of our main problem: Search for Siamese color graphs. Here again we brieﬂy discuss predecessors, particularly the paper [23]. Table 2 on page 83 of this paper depicts results of the enumeration of all amorphic association schemes on 16 points, and a portion of those on 25 points. Computer techniques used for this enumeration are quite close in spirit to our Strategy C of Sect. 9.5.3.3, however without evident use of double cosets. We confess that in the case of this general class of problems there has been no essential progress made in the last two decades. At present, it would seem that such progress depends heavily on the emergence of some new and clever ideas in computer algebra.

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10.2 Implementation and Logistics 10.2.1 Interplay Between Computer Packages As we mentioned, our main tools in this project were COCO and GAP (together with its share packages). Oddly, we did not use Discreta but the reasons were purely logistical: We faced essential diﬃculties in attempting to install it on our existing machines. Ultimately, the author SR wrote his own version of the Kramer–Mesner method in GAP in order to search for Siamese Steiner designs on 40 points. Thus, in this extended sense, Discreta also had an indirect presence in our project. One of the logistical diﬃculties we faced was the incompatibility of data formats used by GAP and COCO. Speciﬁcally, the output of one package had to be constantly adjusted to the format style of the other package prior to data input. Unfortunately, this problem seems to be of growing severity, particularly as computer algebra packages continue to become more diverse and proliﬁc over time. 10.2.2 Future Incarnations of COCO The package COCO was created more than a decade ago, however its philosophical principles go back to an even earlier time. Nowadays, there is strong motivation to replace it with a more modern version which is wider in scope and more powerful in utility. Preferably, such a version should also be available as a share package of GAP. In fact, some eﬀorts in this direction have already been undertaken by the authors MK, SR and a group of their colleagues, mostly at the level of conceptualization and design. Moreover, certain experimental programs have actually been written in GAP which perform functions of this future “COCO II.” Hopefully, support and promotion of this project will one day result in the creation of a package which will have applicability to a wide audience of researchers in the ﬁeld of algebraic combinatorics. 10.3 Siamese Reﬂections 10.3.1 Weighing Matrices Here we discuss the methods used in [35] to construct the ﬁrst inﬁnite series of color graphs of order q 3 + q 2 + q + 1 and Siamese rank q + 1, q any prime. These methods are based on so-called balanced generalized weighing matrices, in the spirit of an approach developed by Y. Ionin [32]. (See also [21] for further information on weighing matrices.) In [35], the authors start with a weighing matrix

Siamese Combinatorial Objects via Computer Algebra Experimentation

⎛

0

⎜ ⎜I ⎜ ⎜ W =⎜ ⎜I ⎜ ⎜I ⎝ I

I

I

I

0

I

U

I

0

U2

U

U2

0

U2

U

I

I

107

⎞

⎟ U2 ⎟ ⎟ ⎟ U ⎟ ⎟, ⎟ I ⎟ ⎠ 0

and interpret each entry of W as a certain 3 × 3 matrix. From this, they obtain the adjacency matrix of a color graph on 15 vertices. In some sense, W can be viewed as the ‘DNA’ of this color graph, incorporating all of its properties into a convenient, compact form. We checked by computer that, up to isomorphism, this is the same color graph as the one which admits the ﬁrst member of our inﬁnite series of classical Siamese association schemes. In fact, the members of the inﬁnite series constructed by Kharaghani and Torabi have rank q + 1 (q a prime power), and parameters v = q 3 + q 2 + q + 1,

k = q 2 + q,

λ = q − 1,

μ = q + 1,

σ = q + 1.

Using a computer, we veriﬁed for q = 2, 3, 4, 5, 7 that members of this series are isomorphic to our respective classical Siamese color graphs on 15, 40, 85, 156 and 400 points. Indeed, it seems that our series and the one of Kharaghani and Torabi are in fact the same, up to isomorphism. We strongly believe that further computer algebra experimentation with weighing matrices, followed by a subsequent theoretical comprehension, is a task of essential importance. Hopefully, such activity will shed new light on the origins of Siamese objects. 10.3.2 Vertex Transitivity Recall that there exist two non-classical vertex-transitive Siamese color graphs on 40 vertices, one of which is circulant. Note also that every classical Siamese color graph is circulant. This leads to the following intriguing question: Are there any other circulant (or at least vertex-transitive) Siamese color graphs besides the classical ones and those on 40 vertices? 10.3.3 Group N ∼ = (S5 × S3 )+ Group N , which was discussed by us in Sect. 7, deserves special attention for a variety of reasons. We mention just a couple. Enumeration of transitive permutation groups has a very long history, see [61] and [14]. In particular, all transitive permutation groups of degree at most 14 were classiﬁed during the 19th century. The problem of classifying all imprimitive permutation groups of degree 15 was only considered in [46], and a bit later in [44]. According to [46] there are 70 such groups, while in [44] 98 groups were found. In [11] G. Butler, who it seems may have been only partially aware of the results of his predecessors, described all transitive permutation groups

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Table 8. Siamese property versus existence of resolutions and maximal arcs Design #1 #6 #7 #8 #9 [25] [50]

Group order 12130560 288 10368 1296 144 39 13

Spread orbits 2 45 10 16 21 13 One spread

Resolution orbits Many 7217 At least 91838 1232 36 One resolution –

Max arcs – – – – – 3 3

Siamese Yes Yes Yes Yes Yes No No

of degree 15 with the aid of a computer. Among these, he found exactly 98 imprimitive ones, a result that was later conﬁrmed in [14]. To our knowledge, no one has ever investigated the list of H. W. Kuhn [44] to make sure that the groups appearing there are pairwise dissimilar, and thus in agreement with the list of Butler. Nonetheless, we can aﬃrm that group N was deﬁnitely known to Kuhn. Group N , as a particular case (q = 2) of the automorphism groups of regular spreads in P G(3, q), also appears in the context of the famous Kirkman Problem, see Corollary 2 on page 73 of [30]. 10.3.4 Empirical Observations Among the 80 Steiner designs on 15 points, there are four resolvable designs which altogether contain seven non-isomorphic Kirkman systems, see [48]. According to the numeration given in [48], these designs are STS #1, STS #7, STS #19 and STS #61, with respective automorphism groups of order 20160, 288, 12 and 21. Among these four, only #1 and #7 are Siamese, and these two are also “extremal” in the sense of having the greatest number of spreads (56 and 32, respectively). A lower bound on the number of Steiner designs on 40 points is evaluated to be about one million. Nevertheless, for a long time only one such design was known to be resolvable, namely the classical P G(3, 3). Recently, however, M. Greig and A. Rosa found a non-classical example [25] which has three maximal arcs. One more example, also with maximal arcs, appears in [50]. This prompted us to arrange a comparison between a few of our Siamese designs on 40 points and these two newly discovered ones, which we also regard as exceptional. Our ﬁrst impression is that the Siamese property seems to be in direct correlation with “abundance of resolutions,” however it is inversely correlated to the existence of maximal arcs, see Table 8 (the designations #n carry over from Table 7). We intend to investigate these speculations after the enumeration of all Siamese objects on 40 points has been completed.

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Acknowledgments An essential part of this research was conducted while M.K. was a visitor, and S.R. was a graduate student, at the University of Delaware. These authors express their gratitude to that institution. All authors are grateful to S.-Y. Song and his colleagues for comments which contributed to improving the quality of the paper. Finally, it is our pleasure to acknowledge Bruno Buchberger and the coordinators of the Special Semester on Groebner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria. Their collective interest in tutorials which cover algorithmic aspects of algebra and combinatorics mirrors our own, and is most appreciated.

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33. A. A. Ivanov, Construction of some new automorphic graphs using computers, in Air Physics and Applied Mathematics (Moscow) MPhTI, pp. 144– 146, 1981 (in Russian). 34. A. Kerber and R. Laue, Group actions, double cosets and homomorphisms: Unifying concepts for the constructive theory of discrete structures, in M. Hazewinkel, A. A. Ivanov, and A. J. Woldar (eds.) Algebra and Combinatorics: Interactions and Applications. Acta Applicandae Mathematicae, Vol. 52 (1–3), pp. 63–90, Kluwer Academic, Dordrecht, 1998. 35. H. Kharaghani and R. Torabi, On a decomposition of complete graphs, Graphs Comb., 19 (2003), 519–526. 36. M. H. Klin, M. Meszka, S. Reichard, A. Rosa, and E. Spence, The smallest non-rank 3 strongly regular graphs which satisfy the 4-vertex condition, Bayreuther Mathematische Schriften, 74 (2005), 145–205. 37. M. Ch. Klin, R. P¨ oschel, and K. Rosenbaum, Angewandte Algebra, Vieweg, Braunschweig, 1988. 38. M. H. Klin and S. Reichard, Siamese association schemes in miniature: A link between generalized quadrangles, graphs and Steiner designs. Abstracts of AMS Meeting (Bloomington), 2003, 985-05-232. 39. M. H. Klin, R. Reichard, and A. J. Woldar, Siamese objects and their relation to color graphs, association schemes and Steiner designs, Bull. Belgian Math. Soc. (5), 12 (2005), 845–857. 40. M. H. Klin, S. Reichard, and A. J. Woldar, Siamese association schemes and Siamese Steiner designs. I. Introduction (in preparation). 41. M. H. Klin, S. Reichard, and A. J. Woldar, Siamese association schemes and Siamese Steiner designs. II. Objects on 15 points (in preparation). 42. M. Klin, C. R¨ ucker, G. R¨ ucker, and G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras. MATCH, 40 (1999), 7–138. 43. E. S. Kramer and D. M. Mesner, t-Designs on hypergraphs, Discrete Math., 15 (1976), 263–296. 44. H. W. Kuhn, On imprimitive substitution groups, Am. J. Math., 26 (1904), 45–102. 45. R. Laue, Computing double coset representatives for the generation of solvable groups, Lect. Notes Comput. Sci., 144 (1982), 65–70. 46. E. N. Martin, On the imprimitive substitution groups of degree ﬁfteen and the primitive substitution groups of degree eighteen, Am. J. Math., 23 (1901), 259–286. 47. R. Mathon, Lower bounds for Ramsey numbers and association schemes, J. Comb. Theory, Ser. B, 42 (1987), 122–127. 48. R. A. Mathon, K. T. Phelps, and A. Rosa, Small Steiner triple systems and their properties, Ars Combinatoria, 15 (1983), 3–110. 49. B. D. McKay, nauty User’s Guide (Version 1.5), Technical Report TRCS-90-02, Computer Science Department, Australian National University, 1990.

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50. S. Milici, A. Rosa, and V. Voloshin, Colouring Steiner systems with speciﬁed block colour patterns, Discrete Math. (1–3), 240 (2001), 145–160. 51. E. H. Moore, Tactical memoranda I–III, Am. J. Math., 18 (1896), 264– 303. 52. S. E. Payne, All generalized quadrangles of order 3 are known, J. Comb. Theory, 18 (1975), 203–206. 53. S. E. Payne, Tight point sets in ﬁnite generalized quadrangles, Congressus Numerantium, 60 (1987), 243–260. 54. S. E. Payne, and J. A. Thas, Finite Generalized Quadrangles, Research Notes in Mathematics, Pitman, London, 1984. 55. S. Reichard, Computational and Theoretical Analysis of Coherent Conﬁgurations and Related Incidence Structures, Thesis, University of Delaware, Newark, 2003. ¨ 56. R. Remak, Uber Untergruppen direkter Produkte von drei Faktoren, Crelle’s J. Reine Angewandte Math., 166 (1931), 65–100. 57. B. Schmalz, t-Designs zu vorgegebener Automorphismengruppe, Thesis, Universit¨ at Bayreuth, Bayreuther Mathematische Schriften, 41 (1992), 1– 164. 58. B. Schmalz, The t-designs with prescribed automorphism group, new simple 6-designs, J. Comb. Des., 1 (1993), 126–170. 59. M. Sch¨onert et al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D f¨ ur Mathematik, Rheinisch-Westf¨alische Technische Hochschule, Aachen, 1995. 60. W. R. Scott, Group Theory, Dover, New York, 1987. 61. M. W. Short, The Primitive Soluble Permutation Groups of Degree Less than 256, Lecture Notes in Mathematics, Vol. 1519, Springer, Berlin, 1992. 62. L. H. Soicher, GRAPE: A system for computing with graphs and groups, in L. Finkelstein and W. M. Kantor (eds.) Groups and Computation, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 11, pp. 287–291, Amer. Math. Soc., Providence, 1993. 63. E. Spence, http://www.maths.gla.ac.uk/∼es/srgraphs.html. 64. J. J. Sylvester, Elementary researches in the analysis of combinatorial aggregation, Philos. Mag., 24 (1844), 285–296. 65. J. A. Thas and S. E. Payne, Spreads and ovoids in ﬁnite generalized quadrangles, Geometrica Dedicata, 52 (1994), 227–253. 66. J. A. Todd, As it might have been, Bull. London Math. Soc., 2 (1970), 1–4. 67. A. Wasserman, Finding simple t-designs with enumeration techniques, J. Comb. Des. (2), 6 (1998), 79–90.

Using Gr¨ obner Bases to Investigate Flag Algebras and Association Scheme Fusion Douglas A. Leonard Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA. [email protected]

Summary. This paper is meant primarily as a tutorial on how to phrase problems in association schemes in the language of Gr¨ obner bases and use the computational results provided by those bases, though it does contain fusion scheme computations not previously found in the literature.

Key words: Gr¨ obner bases, Flag algebras, Association schemes

1 Introduction It is instructive to see how the use of Gr¨ obner bases [2] can clarify certain computational and theoretic aspects of various topics. Here the choice of topics come from association schemes and fusion of ﬂag algebras arising from generalized n-gons and 2-(v, k, 1) designs [6]. Three diﬀerent ways of using concepts related to Gr¨obner bases are given as a guide for using such concepts in similar types of problems. Often it is the case that combinatorialists have to sort out some form of truth about a given object based on its parameters. This can translate into pages and pages of manipulations of intermediate results involving polynomial equations, with the attendant derivations and justiﬁcations. But these problems can usually be easily rephrased in terms of ﬁnding common zeros of a system of polynomial equations, and knowing for what values of the parameters this happens. This is exactly what Gr¨obner bases are meant to accomplish. So Gr¨ obner basis theory can provide direct computational answers without the need for further justiﬁcation, once problems are phrased in terms of generators (polynomials that should be zero) for an ideal in a multivariate polynomial ring, and the need to know the corresponding varieties (set of common zeros). This should be useful either to those wishing to use Gr¨ obner bases to clarify and/or simplify computations or to those interested in seeing how Gr¨ obner bases can be used in “applied theoretical” settings.

M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 3,

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And it could conceivably use the same system of equations to produce results overlooked in some ad hoc approach to solving them. Now consider the concrete problem of investigating a parameterized series of association schemes, describing the structure constants, determining fusion partitions, and investigating those fusions found. Much of this work was done by the Soviet school for metric (one parameter) schemes (see, for example, the survey [4] and http://www.ricam.oeaw.ac.at/specsem/SRS/groeb/download/ Muzychuk.pdf). See also [9]. The next natural stage is to investigate dihedral (two parameter) schemes in the terminology of Zieschang, coming mainly from ﬂag algebras, which were introduced by a number of authors [5, 10, 7, 11, 13]. The paper [6], which started this investigation, used ad hoc methods to do the computations. So here those methods have been replaced by the systematic Gr¨ obner basis methods, which make the actual computations invisible to the reader. There are new computations relative to fusion expanding on this previous work, with the MAGMA [8] code that generated them, to whet the appetite of the audience.

2 Gr¨ obner Basis Preliminaries There are many good books covering this material, with [3] being the ﬁrst author’s favorite. The following is a brief introduction to ideals and varieties in polynomial rings for those unfamiliar with the topic. Given some variables xn , . . . , x1 and a coeﬃcient ring R, R[xn , . . . , x1 ] denotes the ring of (ﬁnite) R-linear combinations of (ﬁnite) products in these variables, and seems to be called either a multivariate polynomial ring or a free(associative)algebra, depending on whether the variables commute with each other or not. Even to be able to write down polynomials in a canonical way, it is necessary to have a monomial order (a total order with obvious extra properties). The (default) lexicogrphical monomial order is based on comparing products by considering their indices (that is, exponents) lexicographically. So, for instance, in the multivariate polynomial ring R[x3 , x2 , x1 ] the order looks like 1 ≺ x1 ≺ x21 ≺ · · · ≺ x2 ≺ x2 x1 ≺ · · · ≺ x22 ≺ · · · ≺ x3 ≺ · · · which can be described by xi33 xi22 xi11 xj33 xj22 xj11 iﬀ (i3 > j3 ) or (i3 = j3 and i2 > j2 ) or (i3 = j3 and i2 = j2 and i1 > j1 ). The generic total degree orders are the grevlex and glex orders (short for graded reverse lexicographical and graded lexicographical), in which total degree is the ﬁrst concern. These give orders that look like 1 ≺ x1 ≺ x2 ≺ x3 ≺ x21 ≺ x2 x1 ≺ x3 x1 ≺ x22 ≺ · · · and 1 ≺ x1 ≺ x2 ≺ x3 ≺ x21 ≺ x2 x1 ≺ x22 ≺ x3 x1 ≺ · · ·

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respectively. These can be deﬁned in ways similar to those above; but the non-singular matrices ⎛ ⎞ ⎛ ⎞ 111 111 Agrevlex := ⎝1 1 0⎠ Aglex := ⎝1 0 0⎠ 100 010 can be used to reduce these to the lexicographical case by converting the column vector of exponents ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ i3 i3 + i2 + i1 i3 + i2 + i1 ⎝i2 ⎠ to ⎝ i3 + i2 ⎠ or ⎝ ⎠ i3 i1 i3 i2 respectively. The leading monomial of f , denoted here by LM (f ) is the largest monomial occurring in f , relative to the given order. Its coeﬃcient is called the leading coeﬃcient, denoted by LC(f ), and the product LC(f )LM (f ) =: LT (f ) is called the leading term. (Warning: Some authors interchange the words term and monomial, and there is no uniformity of notation in the literature.) An ideal I (two-sided in the case of free algebras) is a subset of a ring closed under addition of its elements and under multiplication by ring elements. Ideals describing ﬁnitely-presented algebras (that is, free associative algebras modulo said ideals) are commonly given in terms of a (ﬁnite) set of generators (referred to as relations when the variables are called generators!) I := g1 , g2 , . . . , gm , so that I consists of all (ﬁnite) R-linear combinations of g1 , . . . , gm . As with vector spaces, where arbitrary generating sets are not as useful as maximal linearly independent sets, called (vector space) bases, it should be no surprise that arbitrary generating sets for ideals (which are unfortunately called bases in some of the literature) are not as important or useful as Gr¨ obner bases, those generating sets B relative to which a “canonical” remainder NormalForm(f, B) for f after division by the elements of B (in any order) can be deﬁned. A more useful computational deﬁnition of Gr¨obner bases is in terms of SPolynomials. If h is a least common multiple of LM (f ) and LM (g), then SPoly(f, g) := LTh(f ) f − g LTh(g) (with suitable generalization in the noncommutative case). Then B is a Gr¨ obner basis if SPoly(f, g) reduces to 0 after division by the elements of B for all choices of f, g ∈ B. In fact this is the foundation for all forms of the Buchberger algorithm; namely that given any generating set, it is possible to produce a Gr¨ obner basis by computing SPolynomials, reducing them by division relative to the current generating set, and appending the results to the generating set if necessary, possibly reducing the other elements along the way, until the set is closed under this combined operation of computing SPolynomials and reducing them. If one has a collection of equations of the form fi = hi that can be written in polynomial form in terms of some (not necessarily polynomial) variables,

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then it is possible to use gi := fi −hi as generators for an ideal in a polynomial ring. Common solutions to the collection of equations then become common zeros of the elements of the ideal I or equivalently of the elements of the Gr¨ obner basis B. These common zeros are called points of the variety V(I) := {v : g(v) = 0 for all g ∈ I}. This variety is generally computed by ﬁnding a Gr¨obner basis relative to some easy ordering, changing it to a (factored) Gr¨ obner basis relative to a lex order, and recursively computing coordinates of the elements of the variety. The importance of all this to a ﬁeld such as algebraic combinatorics is that in the study of a collection of objects relative to a set of parameters, it is typical that questions may be phrased in terms of what parameter values are necessary or suﬃcient to guarantee the feasibility or existence of such objects. Then equations satisﬁed by the parameters can be turned into generators for an ideal for which the variety gives such parameter values. That means that rather than haphazardly sorting through a collection of equations, looking for results about the parameters, it is possible to merely compute a Gr¨ obner basis and a variety, letting the Gr¨obner basis theory replace all the intermediate results, derivations, and justiﬁcations. Sometimes there are unexpected added beneﬁts beyond this, in that proper phrasing of problems in terms of Gr¨obner bases may give insight into the combinatorial structures, deﬁnitions, or other aspects of a problem. This will be exempliﬁed below by considering generalized n-gons and 2-(v, k, 1) designs.

3 Algebraic Combinatorics Preliminaries A matrix algebra W over the ﬁeld C of complex numbers is called a coherent algebra iﬀ the identity matrix, I, is in W , the all-ones matrix, J is in W , the transpose, AT of A is in W , if A is, and the entry-wise products of elements of W are in W . It is well known that each coherent algebra W contains an unique basis of (0, 1) matrices {A0 , A2 , . . . , Ar−1 } such that A0 + A1 + · · · + Ar−1 = J and ATi = Ai . The use of this standard basis can be used to reformulate the notion of a coherent algebra in terms of relations and their graphs, called coherent conﬁgurations. And if A0 = I, these are called association schemes. Many signiﬁcant examples of classes of combinatorial objects may be reformulated in terms of coherent conﬁgurations or association schemes, a few being discussed below. The fact that W is an algebra implies the existence r of (non-negative integer) structure constants pi,j,k such that Ai Aj = k=1 pi,j,k Ak . If the Ai ’s are treated as variables instead of (0, 1) matrices, the algebras are called table algebras [1]. While in many cases it is crucially important to keep track of all combinatorial information about a coherent algebra W , many signiﬁcant feasible conditions are obtained by viewing it merely as a table algebra. In particular, a question about the existence of coherent subalgebras of a given coherent algebra (fusions) may be solvable based only on the knowledge of

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the structure constants. Classes of table algebras may be deﬁned in terms of parameters and relations among those parameters, which suggests applying Gr¨ obner basis techniques to produce feasibility conditions.

4 Deﬁnitions Start with ﬁnite sets V, of v vertices, and B, of b blocks. The basic combinatorial structure called a 1-design is merely a subset D of V × B such that r := #{l ∈ B : (p, l) ∈ D} is independent of p ∈ V and k := #{p ∈ B : (p, l) ∈ D} is independent of l ∈ V. Elements (p, l) ∈ D are called ﬂags; and there are vr = bk such. A 1-design such that any two vertices are incident with at most one block and any two blocks are incident with at most one vertex is called a partial linear space. (Geometers tend to use the word point in place of vertex, the word line in place of block, and the parameters s := k − 1 and t := r − 1 instead of k and r respectively.) Normally the study of vertices is linked to the study of the (symmetric) (0, 1) adjacency matrix, A, with A(pi , pj ) = 1 if and only if

(pi , l), (pj , l) ∈ D

for some l ∈ B.

But here the focus is initially on ﬂags; so let L denote the (symmetric) (0, 1) collinearity matrix indexed by the ﬂags, with L((p1 , l1 ), (p2 , l2 )) = 1 if and only if l1 = l2 , p1 = p2 ; and let N denote the (symmetric) (0, 1) concurrence matrix indexed by the ﬂags, with N ((p1 , l1 ), (p2 , l2 )) = 1 if and only if

p1 = p2 , l1 = l2 .

These matrices already satisfy the conditions L2 = (s − 1)L + sI,

N 2 = (t − 1)N + tI

(with I denoting the vr × vr identity matrix). Generalized n-gons were ﬁrst introduced by Tits [12] in 1959. Standard deﬁnitions are in terms of distance and adjacency, and are geometric in ﬂavor, descriptive of the fact that each point should be contained in an n-gon and in no smaller polygon. In terms of ﬂags and the matrices L and N above this translates into the following matrix-theoretic deﬁnition, which obscures the geometric origin. Let A4i+1 := L(N L)i , A4i := (N L)i , A4i+2 := N (LN )i , A4i+3 := (LN )i+1 , denote the various ﬂag adjacency matrices (a possibly unexpected beneﬁt of this non-geometric approach). Then D will be called a generalized n-gon

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if (Aj , 0 ≤ j ≤ 2n − 1) is a linearly independent set over Z, and hence a basis for the Z-module it generates; but A2n = A2n−1 . (It is a straightforward, instructive exercise to see that this really corresponds to the geometric concept above.) Since A0 = I, AT4i+1 = A4i+1 , AT4i+2 = A4i+2 , AT4i+3 = A4i+4 , and 2n−1 j=0 Aj = J, the all 1’s matrix; the ordered set (Aj : 0 ≤ j ≤ 2n − 1) is a basis for the Z-module it generates, lacking only a multiplication to be the adjacency algebra of an association scheme. If the two matrices L and N are not known, and the parameters t and s are not known either, then it is possible to study generalized n-gons by starting with the coeﬃcient ring Z[t, s] (with a lex monomial order t s), and then a free (associative) algebra Z[t, s][y, x] in two non-commutative variables (with a total degree monomial order with y x). The former means that when writing polynomials in the variables t and s, ti sj should be treated as larger than tk sl if either (i > k) or (i = k and j > l) (and that one shouldn’t write sj ti ). The latter means that when writing polynomials in the (non-commuting) variables y and x monomials, a string such as yxxyx should be treated as larger than either yxy or yxxxy, the ﬁrst because the total degree is greater, the second because the total degree is the same but at the leftmost diﬀerence in the strings y x. (One of the ﬁrst important lessons one learns when working with multivariate polynomial rings is that a proper choice of monomial order is perhaps the most critical step in any problem, and the step most easily ignored completely by those not used to this area.) Then this should be made into a ﬁnitely-presented algebra (quotient ring in two non-commuting variables, representing the two non-commuting matrices N and L respectively) Z[t, s][y, x]/f1 , f2 , f3 with f1 := x2 − (s − 1)x − s, f2 := y 2 − (t − 1)y − t, and f3 := (yx)l − (xy)l if n = 2l or f3 := y(xy)l − (xy)l x if n = 2l + 1. The import here is that the ideal I := f1 , f2 , f3 consists of all those polynomials that are supposed to be equal to zero under the given assumptions. (Such a (table) algebra, depending only on the structure constants and not on some (0, 1) matrices, may exist whether or not a corresponding n-gon does. At this stage it is not even necessary to assume knowledge of classical necessary conditions for the existence of such n-gons.)

5 An Application of SPolynomials and Reduction Buchberger’s algorithm for computing Gr¨ obner bases for ideals (such as f1 , f2 , f3 above) is based on the central observation above that bases (as opposed to generating sets) for ideals should be closed under the operation of reduction of SPolynomials. Consider an example of this type of computation for the n-gon case with n = 2l + 1. Since LM (yf3 ) = y 2 (xy)l = LM (f2 (xy)l ), the corresponding (non-commutative) SPolynomial would be

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SPolynomial (f3 , f2 ) := yf3 − f2 (xy)l = −y(xy)l x + (t − 1)y(xy)l + t(xy)l , which would then be reduced to a remainder (yf3 − f2 (xy)l ) + f3 x + (xy)l f1 − (t − 1)f3 = (t − s) (xy)l x + (xy)l using the explicit division by the set {f1 , f2 , f3 } given. Thus a single SPolynomial computation, and the use of the assumption that (xy)l x and (xy)l are linearly independent, already forces t − s = 0. So it follows that: Proposition 1. Generalized (2l + 1)-gons with parameter pairs (t, s) can only exist if t = s. Remark. Again, this can be viewed as a result about table algebras.

6 Structure Constants of Association Schemes In light of the proposition above, consider only generalized 2l-gons in their ﬁnitely-presented algebra form. If a total degree monomial order (with ai aj for i > j) is used, then a minimal, reduced Gr¨ obner basis (that is, one with a minimal number of elements, and no leading monomial of one dividing any monomial of another) for the ideal of relations, I, will have all its elements of the form ai aj − NormalForm(ai aj , I) with NormalForm(ai aj , I) =

2n−1

pi,j,k ak ,

k=0

describing the multiplication in the algebra, as well as determining the structure constants pi,j,k that make this the adjacency algebra of an association scheme. (If this is not immediately obvious, see the proof below.) Thus a simple Gr¨obner basis computation relative to an appropriate total degree monomial order gives constructively the following result (with concrete examples below): Proposition 2. Z[t, s][a4l−1 , . . . , a1 ]/I with I the ideal of relations with generating set containing the basis relations from the deﬁnition of a generalized 2l-gon: f1 := a21 − (s − 1)a1 − s, f2 := a22 − (t − 1)a2 − t, f3 := (a2 a1 )l − (a1 a2 )l ;

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together with the relations gotten from the deﬁnitions of the aj ’s: a1 (a2 a1 )i − a4i+1 , (a2 a1 )i − a4i , (a1 a2 )i+1 − a4i+3 , 0 ≤ i < l,

a2 (a1 a2 )i − a4i+2 ,

corresponds to the adjacency algebra of an association scheme. (Note that a0 = 1 is implicit in the calculations, and sometimes explicit in the theory.) Moreover, if a total degree monomial order (with ai aj for i > j) is used, then a minimal, reduced Gr¨ obner basis for I will have all its elements of the form ai aj − NormalForm(ai aj , I) with NormalForm(ai aj , I) =

4l−1

pi,j,k ak ,

k=0

which corresponds to a complete description of the algebra multiplication and a determination of the structure constants pi,j,k . Proof. Given that the ai are linearly independent, there can be no relations with leading monomial of degree 1. Given that the ai ’s form an algebra, ai aj must be expressible as a linear combination k pi,j,k ak , since all elements of 4l−1 the algebra are of this form. Hence ai aj − k=0 pi,j,k ak must be basis elements in any total degree monomial order Gr¨ obner basis. And any monomial of total degree greater than 2 is divisible by one of degree 2, so can’t be a leading monomial in any minimal total degree Gr¨ obner basis.

7 Fusion Now consider a partition Π of the set {0, . . . , 4l − 1}, and the corresponding fusion of classes (sum of their respective matrices) {Ak : k ∈ γ}, γ ∈ Π. Bγ := This could conceivably determine a fusion scheme; that is, be an association scheme with fewer classes than the original, if {0} is a part, γ := {k : T k ∈ γ, A k = A k } is a part for each γ ∈ Π, and the structure constant Pα,β,γ := i∈α j∈β pi,j,k is independent of the choice of k ∈ γ, for all parts α, β, and γ in Π. This can be viewed as another Gr¨ obner basis problem, but this time in the multivariate polynomial coeﬃcient ring Z[t, s] (with two commuting variables t and s, the integer parameters). The ideal in question is generated by the relations (forced by the third item above): pi,j,k1 − pi,j,k2 i∈α j∈β

i∈α j∈β

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for all α, β, and γ in Π and all k1 , k2 ∈ γ, provided Π is a good partition (that is, satisfying the ﬁrst two items above). (This is a slightly diﬀerent condition than that of good sets in [6].) Then the variety of this ideal determines all possible parameter pairs (t, s) for which the good partition Π produces a fusion scheme. Of course, it is necessary to restrict the variety to pairs of positive integers, and probably to ignore the trivial case t = 1 = s in which the original generalized 2l-gon is merely a 2l-gon, and fusion corresponds to Schur rings over the dihedral group of order 2l. The varieties involved are computed most easily using a lexicographical monomial order on Z[t, s] and a factored, minimal, reduced Gr¨ obner basis.

8 2-(v, k, 1) Designs Consider a similar problem of ﬂag adjacency matrices for 2-(v, k, 1) designs (also called Steiner systems S(2, k, v) or just 2-designs); that is, 1-designs for which every 2 vertices determine an unique block. The matrices for this are similar in ﬂavor to those for the generalized n-gons: A0 := I, A1 := L, A2 := N, A3 := LN, A4 := N L, A5 := LN L, A6 := N LN − LN L with LN LN = sA6 + (s − 1)A5 + sA4 ,

N LN L = sA6 + (s − 1)A5 + sA3 .

Since A0 = I, AT1 = A1 , AT2 = A2 , AT3 = A4 , AT5 = A5 , AT6 = A6 , and 6 j=0 Aj = J, the all 1’s matrix; the ordered set (Aj : 0 ≤ j ≤ 6) is a basis for the Z-module it generates, lacking only a multiplication to be the adjacency algebra of an association scheme. Then this should be made into a ﬁnitely-presented algebra as before Z[t, s][y, x]/f1 , f2 , f3 , f4 with f1 := x2 − (s − 1)x − s, f2 := y 2 − (t − 1)y − t, and f3 := (yx)2 − s(yxy − xyx) − (s − 1)xyx) − sxy), f4 := (xy)2 − s(yxy − xyx) − (s − 1)xyx − syx. Remark. When s < t, A6 = 0, so this is a rank 7 association scheme. However, if s = t, then A6 = 0 and the 2-(v, k, 1) design is a projective plane or equivalently a generalized 3-gon, covered earlier.

9 Code and Output There are also technical diﬃculties to be considered when using existing computer algebra packages. For instance, in MAGMA, the coeﬃcient ring of a

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ﬁnitely-presented algebra needs to be a ﬁeld. So it is necessary to use Q(t, s), the function ﬁeld in two variables over the rational ﬁeld Q, with a total degree monomial order, and then map results to Q[t, s] with the desired lex monomial order, where factorization can be done. The MAGMA code below has been automated so that the only parameter input necessary is l (meaning n/2). The output is a list of partitions giving rise to fusion schemes, together with a factored Gr¨obner basis from which it is relatively easy to read the corresponding parameter pairs (t, s), if such positive integer pairs exist. Although the computations are (for better or worse) no longer visible to the reader, it is instructive to have some idea of what is happening. As an example from the generalized quadrangle case, the partition Π := [[1, 2, 3, 4, 7][5, 6]] could only give a fusion scheme if ts2 − 2ts, t2 − 2t − s2 + 2s, t2 s − ts2 , and t2 − ts2 + 3ts − 4t + s2 − 4s + 4 (gotten from the fusion condition 3 above) are all zero. An interreduction of these generators already gives a Gr¨ obner basis (t2 − 2t, ts − 2t, s2 − 2s) for the ideal they generate. And a recursive calculation of s and t gives s = 0, t = 0, s = 2, t = 0, or s = 2, t = 2. Of these 3 rational points in the variety, clearly only (t, s) = (2, 2) is useful in this fusion context. (Currently the code actually produces a modiﬁed Gr¨ obner basis (t − 2, s − 2) by removing factors such as t and s which can’t ever lead to positive integer solutions.) There are examples for which the solutions must be done by hand. For instance, in partition 48 in the hexagon output below, one factor is T 2 − T S + 2T + 1 which gives a one dimensional variety s = (t + 1)2 /t over the rationals, but only the single positive integer solution s = 4, t = 1. The code is written to search for a good partition, compute diﬀerences of coeﬃcients that should be equal for fusion to occur, ﬁnd a Gr¨ obner basis for ideal of all such diﬀerences, and output same, at least in the cases in which there might conceivably be an element in the variety with all coordinates positive. In most cases, it is relatively easy to read oﬀ any parameter pairs with both entries positive integers, and to ignore those with only the trivial solution t = 1 = s. Note that the code produces a Gr¨ obner basis describing the original adjacency algebra before tackling the fusion problem (an expected beneﬁt); so it is possible to see the structure constants, not in a table or even in the form ai aj = k pi,j,k ak , but in the form ai aj − k pi,j,k ak .

10 Concluding Remarks There are further computations available at the author’s home page: http://www.dms.auburn.edu/∼leonada. Interpretation of the results obtained is of independent interest. Such interpretation for 4-gons and Steiner designs is contained in [6]. But as a striking example of the signiﬁcance of this interpretation step, note that 3-gons with s = t = 4, [4] correspond to a sporadic

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strongly regular graph with v = 105, k = 32, λ = 4, μ = 12. This graph is a subgraph of the famous sporadic McLaughlin graph on 275 vertices. Note also that even though a scheme itself may not exist, some fusion of the table algebra may have a combinatorial interpretation. Acknowledgments It is important to recognize RICAM (Austrian Academy of Sciences), RISC (Johannes Kepler University), and Bruno Buchberger personally for the opportunities provided by the D1 Workshop during the Special Semester on Gr¨ obner bases at Linz, Austria in May, 2006, without which this paper would not have been written. The desirability of such a tutorial paper on the use of Gr¨ obner bases in algebraic combinatorics was suggested by Mikhail Klin, who is also responsible for the underlying topics and much of the historical combinatorial context as well.

Appendix A //common code to search for fusion R:=PolynomialRing(Q,2); co:=function(f,mon) return MonomialCoefficient(f,mon); end function; fusion:=function(J) return &+[A.(N+1-j): j in J]; end function; prod:=function(I,J) return NormalForm(fusion(I)*fusion(J),ID); end function; relations:=function(I,J,K) W:=prod(I,J); return [(co(W,A.(N+1-K[1]))-co(W,A.(N+1-k))) @homR|T,S>: k in K]; end function; RELATIONS:=function(part) rel:=[]; for i in [1..#part] do for j in [1..#part] do for k in [1..#part] do rel:=rel cat relations(part[i],part[j],part[k]); end for; end for; end for; return rel; end function; max:=function(PV,j) if j gt 1 then return Maximum({PV[i]: i in [1..j-1]}); else return -1; end if;

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end function; partno:=0; goodpartno:=0; vector:=[1:i in [1..N]]; v1:=N; while v1 gt 1 do Bound:=max(vector,N+1); B:=[[]: i in [1..Bound]]; for i in [1..N] do Append(~B[vector[i]],i); end for; symmetric:=true; for i in [1..Bound] do if #{vector[AT[j]]: j in B[i]} ne 1 then symmetric:=false; break; end if; end for; if symmetric then partno+:=1; id:=ideal; gb:=GroebnerBasis(id); pos_sol:=true; if #gb ne 0 then for i in [1..#gb] do m:=Minimum(Coefficients(gb[i])); if m gt 0 then pos_sol:=false; break; end if; end for; end if; if pos_sol then goodpartno+:=1; partno, "partition" cat IntegerToString(goodpartno) cat "=",B; if #gb ne 0 then for i in [1..#gb] do Factorization(gb[i]); end for; end if; end if; end if; v2:=v1; bound:=1+max(vector,v1); if vector[v1] ge bound then while vector[v1] ge bound and v1 gt 1 do vector[v1]:=1; v1-:=1; v2:=N; if v1 gt 1 then bound:=1+max(vector,v1); end if; end while; if v1 gt 1 then vector[v1]+:=1; v1:=v2; end if;

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else if v1 gt 1 then vector[v1]+:=1; end if; end if; end while; //preamble code for generalized n-gons l:=2;//the only parameter that needs to be changed from //one run to the next n-gons for n=2l, as rank N+1 //association schemes of flags N:=4*l-1; AT:=[i : i in [1..N]]; for i in [1..l-1] do AT[4*i]:=4*i-1; AT[4*i-1]:=4*i; end for; Q:=RationalField(); FF:=FunctionField(Q,2); A:=FreeAlgebra(FF,N); AssignNames(~A,["a" cat IntegerToString(N+1-i): i in [1..N]]); a1:=A.N;a2:=A.(N-1); rel1:=a1^2-s-(s-1)*a1; rel2:=a2^2-t-(t-1)*a2; rel3:=(a2*a1)^l-(a1*a2)^l; def1:=[(a1*a2)^i-A.(N-4*i+2): i in [1..l]]; def2:=[(a2*a1)^i-A.(N-4*i+1): i in [1..l-1]]; def3:=[a1*(a2*a1)^i-A.(N-4*i): i in [1..l-1]]; def4:=[a2*(a1*a2)^i-A.(N-4*i-1): i in [1..l-1]]; ID:=ideal; G:=GroebnerBasis(ID);G;#G; //Gr\"obner basis for the association scheme of a generalized //quadrangle a7^2 -(t^2s^2-t^2s+t^2-ts^2-t+s^2-s+1)*a7 -(t^2s^2-t^2s-ts^2+ts+s^2-s)*a6 -(t^2s^2-t^2s+t^2-ts^2+ts-t)*a5-(t^2s^2-t^2s-ts^2+ts)*a4 -(t^2s^2-t^2s-ts^2+ts)*a3-(t^2s^2-ts^2)*a2 -(t^2s^2-t^2s)*a1-(t^2s^2)*a0, a7*a6-(t^2s-t^2-ts+t+s-1)*a7-(t^2s-2ts+s)*a6 -(t^2s-t^2-ts+t)*a5-(t^2s-ts)*a4 -(t^2s-ts)*a3-(t^2s)*a1, a7*a5-(ts^2-ts+t-s^2+s-1)*a7-(ts^2-ts-s^2+s)*a6 -(ts^2-2ts+t)*a5-(ts^2-ts)*a4 -(ts^2-ts)*a3-(ts^2)*a2, a7*a4-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a3, a7*a3-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a4, a7*a2-(t-1)*a7-(t)*a5, a7*a1-(s-1)*a7-(s)*a6, a6*a7-(t^2s-t^2-ts+t+s-1)*a7-(t^2s-2ts+s)*a6

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-(t^2s-t^2-ts+t)*a5 -(t^2s-ts)*a4-(t^2s-ts)*a3-(t^2s)*a1, a6^2 -(t^2-2t+1)*a7-(ts-t)*a6-(t^2-t)*a5-(t^2s-ts)*a2 -(t^2s)*a0, a6*a5-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a3, a6*a4-(t-1)*a7-(ts-t)*a4-(ts)*a2, a6*a3-(t-1)*a7-(t)*a5, a6*a2-(t-1)*a6-(t)*a4, a6*a1-(1)*a7, a5*a7-(ts^2-ts+t-s^2+s-1)*a7-(ts^2-ts-s^2+s)*a6 -(ts^2-2ts+t)*a5-(ts^2-ts)*a4-(ts^2-ts)*a3-(ts^2)*a2, a5*a6-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a4, a5^2 -(s^2-2s+1)*a7-(s^2-s)*a6-(ts-s)*a5-(ts^2-ts)*a1-(ts^2), a5*a4-(s-1)*a7-(s)*a6, a5*a3-(s-1)*a7-(ts-s)*a3-(ts)*a1, a5*a2-(1)*a7, a5*a1-(s-1)*a5-(s)*a3, a4*a7-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a3, a4*a6-(t-1)*a7-(t)*a5, a4*a5-(s-1)*a7-(ts-s)*a4-(ts)*a1, a4^2 -(1)*a7, a4*a3-(s-1)*a6-(ts-s)*a2-(ts)*a0, a4*a2-(1)*a6, a4*a1-(s-1)*a4-(s)*a2, a3*a7-(ts-t-s+1)*a7-(ts-s)*a6-(ts-t)*a5-(ts)*a4, a3*a6-(t-1)*a7-(ts-t)*a3-(ts)*a2, a3*a5-(s-1)*a7-(s)*a6, a3*a4-(t-1)*a5-(ts-t)*a1-(ts)*a0, a3^2 -(1)*a7, a3*a2-(t-1)*a3-(t)*a1, a3*a1-(1)*a5, a2*a7-(t-1)*a7-(t)*a5, a2*a6-(t-1)*a6-(t)*a3, a2*a5-(1)*a7, a2*a4-(t-1)*a4-(t)*a1, a2*a3-(1)*a6, a2^2 -(t-1)*a2-(t)*a0, a2*a1-(1)*a4, a1*a7-(s-1)*a7-(s)*a6, a1*a6-(1)*a7, a1*a5-(s-1)*a5-(s)*a4, a1*a4-(1)*a5, a1*a3-(s-1)*a3-(s)*a2, a1*a2-(1)*a3, a1^2 -(s-1)*a1-(s)*a0

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//edited fusion partitions for generalized 4-gons, //eliminating those with only t=1=s //with the Gr\"obner basis for each to describe those pairs //(t,s) for which fusion occurs partition1=[[1,2,3,4,5,6,7]] [] partition3=[[1,2,3,4,5,7],[6]] [T-1] partition4=[[1,2,3,4,6,7],[5]] [S-1] partition5=[[1,2,3,4,7],[5,6]] [T-2,S-2] partition10=[[1,2,7],[3,4],[5,6]] [T-2,S-2] partition11=[[1,2],[3,4],[5,6],[7]] [T-S] partition13=[[1,3,4,5,6,7],[2]] [] partition14=[[1,3,4,6],[2,5,7]] [S-1] partition15=[[1,3,4,6],[2,5],[7]] [S-1] partition17=[[1,3,4,6],[2,7],[5]] [S-1] partition18=[[1,3,4,6],[2],[5,7]] [] partition19=[[1,3,4,6],[2],[5],[7]] [S-1] partition20=[[1,6,7],[2,3,4,5]] [T-1] partition21=[[1,6],[2,3,4,5],[7]] [T-1] partition22=[[1],[2,3,4,5,6,7]] [] partition23=[[1,7],[2,3,4,5],[6]] [T-1] partition24=[[1],[2,3,4,5],[6,7]] [] partition25=[[1],[2,3,4,5],[6],[7]] [T-1] partition27=[[1,5,6],[2],[3,4,7]] [T-1,S^2-4*S+3]=[T-1,(S-3)(S-1)] partition29=[[1,6,7],[2,5],[3,4]] [T-1,S-3] partition30=[[1,6],[2,5,7],[3,4]]

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[T-3,S-1] partition32=[[1],[2,5,6],[3,4,7]] [T^2-4*T+3,S-1]=[(T-3)(T-1),S-1] partition33=[[1],[2,5,7],[3,4,6]] [S-1] partition34=[[1],[2,5],[3,4],[6],[7]] [S-1] partition35=[[1,6,7],[2],[3,4,5]] [T-1] partition36=[[1,6],[2],[3,4],[5],[7]] [T-1] partition38=[[1],[2],[3],[4],[5],[6],[7]] [] //edited fusion partitions for generalized 6-gons, //eliminating those with only t=1=s //with the Gr\"obner basis for each to describe those pairs //(t,s) for which fusion occurs partition1=[[1,2,3,4,5,6,7,8,9,10,11]] [] partition7=[[1,2,3,4,5,7,8,9,10,11],[6]] [T-1] partition8=[[1,2,3,4,6,7,8,9,10,11],[5]] [S-1] partition26=[[1,2],[3,4],[5,6],[7,8],[9,10],[11]] [T-S] partition30=[[1,3,4,5,6,7,8,9,10,11],[2]] [] partition34=[[1,3,4,5,6,7,8,10],[2],[9,11]] [T-S] partition37=[[1,3,4,6,9,11],[2,5,7,8,10]] [S-1] partition41=[[1,3,4,6,9,11],[2,5,10],[7,8 ]] [T^2-5*T+4,S-1]=[(T-1)(T-4),S-1] partition42=[[1,3,4,6,9,11],[2,5],[7,8],[10]] [S-1] partition45=[[1,3,4,6,9,11],[2,7,8,10],[5]] [S-1] partition46=[[1,3,4,6,9,11],[2,7,8],[5,10]] [T-2,S-1] partition47=[[1,3,4,6,9,11],[2,10],[5,7,8]] [T-2,S-1] partition48=[[1,3,4,6,9,11],[2],[5,7,8,10]] [T^2*S-T^2-T*S^2+3*T*S-2*T+S-1]=[(S-1)(T^2-TS+2T+1)] so S=1 or (T=1,S=4)

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partition49=[[1,3,4,6],[2],[5,7,8,10],[9,11]] [] partition54=[[1,3,4,6,9,11],[2],[5],[7],[8],[10]] [S-1] partition55=[[1,3,4,9],[2,7,8,10],[5],[6,11]] [S-1] partition58=[[1,6,7,8,9],[2,3,4,5,10,11]] [T-1] partition62=[[1,6,9],[2,3,4,5,10,11],[7,8]] [T-1,S^2-5*S+4]=[T-1,(S-4)(S-1)] partition63=[[1,6],[2,3,4,5,10,11],[7,8],[9]] [T-1] partition67=[[1],[2,3,4,5,6,7,8,9,10,11]] [] partition69= [[1],[2,3,4,5,6,7,8,9],[10,11]] [T-S] partition72=[[1,7,8,9],[2,3,4,5,10,11],[6]] [T-1] partition73=[[1,7,8],[2,3,4,5,10,11],[6,9]] [T-1,S-2] partition74=[[1,9],[2,3,4,5,10,11],[6,7,8]] [T-1,S-2] partition75=[[1],[2,3,4,5,10,11],[6,7,8,9]] [T^2*S-T*S^2-3*T*S-T+S^2+2*S+1]=[(T-1)(TS-S^2-2S-1)] so T=1 or (S=1,T=4) partition76=[[1],[2,3,4,5],[6,7,8,9],[10,11]] [] partition81=[[1],[2,3,4,5,10,11],[6],[7],[8],[9]] [T-1] partition83=[[1,7,8,9],[2,3,4,10],[5,11],[6]] [T-1] partition100=[[1],[2,5,7,8,10],[3,4,6,9,11]] [S-1] partition105=[[1],[2,5,10],[3,4,11],[6,9],[7,8]] [T^2-5*T+4,S-1]=[(T-4)(T-1),S-1] partition106=[[1],[2,5],[3,4],[6,9],[7,8],[10],[11]] [S-1] partition108=[[1,6,7,8,9],[2],[3,4,5,10,11]] [T-1] partition113=[[1,6,9],[2],[3,4,11],[5,10],[7,8]] [T-1,S^2-5*S+4]=[T-1,(S-4)(S-1)] partition114=[[1,6],[2],[3,4],[5,10],[7,8],[9],[11]] [T-1] partition124=[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]] []

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//edited fusion partitions for generalized 6-gons, //eliminating those with only t=1=s //with the Gr\"obner basis for each to describe those pairs //(t,s) for which fusion occurs partition1=[[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]] [] partition10=[[1,2,3,4,5,7,8,9,10,11,12,13,14,15],[6]] [T-1] partition16=[[1,2,3,4,6,7,8,9,10,11,12,13,14,15],[5]] [S-1] partition40=[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14], [15]] [T-S] partition49=[[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]] [] partition53=[[1,3,4,5,6,7,8,9,10,11,12,14],[2],[13,15]] [T-S] partition55=[[1,3,4,5,6,7,8,10,13,15],[2],[9,11,12,14]] [T-1,S-2] partition56=[[1,3,4,6,9,11,12,14],[2,5,7,8,10,13,15]] [S-1] partition61=[[1,3,4,6,9,11,12,14],[2,5,7,8,15],[10,13]] [T-2,S-1] partition67=[[1,3,4,6,9,11,12,14],[2,5,15],[7,8],[10,13]] [T-2,S-1] partition68=[[1,3,4,6,9,11,12,14],[2,5],[7,8],[10,13], [15]] [S-1] partition73=[[1,3,4,6,9,11,12,14],[2,7,8,10,13,15],[5]] [S-1] partition78=[[1,3,4,6,9,11,12,14],[2,7,8,13],[5],[10,15]] [S-1] partition82=[[1,3,4,6,9,11,12,14],[2],[5,7,8,10,13,15]] [S-1] partition84=[[1,3,4,6,9,11,12,14],[2],[5,7,8,10],[13,15]] [S-1] partition87=[[1,3,4,6,13,15],[2],[5,7,8,10],[9,11,12,14]] [T-1,S-2] partition88=[[1,3,4,6],[2],[5,7,8,10],[9,11,12,14],[13,15]] [] partition100=[[1,3,4,6,9,11,12,14],[2],[5],[7],[8],[10],[13],[15]] [S-1] partition101=[[1,3,4,9],[2,7,8,13],[5],[6,11,12,14],[10,15]] [S-1] partition104=[[1,6,7,8,9,14,15],[2,3,4,5,10,11,12,13]]

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[T-1] partition110=[[1,6,7,8,15],[2,3,4,5,10,11,12,13],[9,14]] [T-1,S-2] partition115=[[1,6,15],[2,3,4,5,10,11,12,13],[7,8],[9,14]] [T-1,S-2] partition116=[[1,6],[2,3,4,5,10,11,12,13],[7,8],[9,14], [15]] [T-1] partition120=[[1],[2,3,4,5,6,7,8,9,10,11,12,13,14,15]] [] partition122=[[1],[2,3,4,5,6,7,8,9,10,11,12,13],[14,15]] [T-S] partition124=[[1],[2,3,4,5,6,7,8,9,14,15],[10,11,12,13]] [T-2,S-1] partition125=[[1,7,8,9,14,15],[2,3,4,5,10,11,12,13],[6]] [T-1] partition130=[[1,7,8,14],[2,3,4,5,10,11,12,13],[6],[9,15]] [T-1] partition134=[[1],[2,3,4,5,10,11,12,13],[6,7,8,9,14,15]] [T-1] partition136=[[1],[2,3,4,5,10,11,12,13],[6,7,8,9],[14,15]] [T-1] partition138=[[1],[2,3,4,5,14,15],[6,7,8,9],[10,11,12,13]] [T-2,S-1] partition139=[[1],[2,3,4,5],[6,7,8,9],[10,11,12,13],[14,15]] [] partition152=[[1],[2,3,4,5,10,11,12,13],[6],[7],[8],[9], [14],[15]] [T-1] partition156=[[1,7,8,14],[2,3,4,10],[5,11,12,13],[6],[9,15]] [T-1] partition183=[[1],[2,5,7,8,10,13,15],[3,4,6,9,11,12,14]] [S-1] partition185=[[1],[2,5,7,8,15],[3,4,6,9,14],[10,13],[11,12]] [T-2,S-1] partition197=[[1],[2,5,15],[3,4,14],[6,9],[7,8],[10,13], [11,12]] [T-2,S-1] partition198=[[1],[2,5],[3,4],[6,9],[7,8],[10,13],[11,12], [14],[15]] [S-1] partition203=[[1,6,7,8,9,14,15],[2],[3,4,5,10,11,12,13]] [T-1] partition205=[[1,6,7,8,15],[2],[3,4,5,10,13],[9,14],[11,12]] [T-1,S-2]

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partition221=[[1,6,15],[2],[3,4,13],[5,10],[7,8],[9,14], [11,12]] [T-1,S-2] partition222=[[1,6],[2],[3,4],[5,10],[7,8],[9,14],[11,12], [13],[15]] [T-1] partition259=[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11], [12],[13],[14],[15]] [] //preamble code for 2-(v,k,1) designs N:=6; AT:=[1,2,4,3,5,6]; Q:=RationalField(); FF:=FunctionField(Q,2); A:=FreeAlgebra(FF,N); AssignNames(~A,["a" cat IntegerToString(N+1-i): i in [1..N]]); a1:=A.6; a2:=A.5; a3:=A.4; a4:=A.3; a5:=A.2; a6:=A.1; rel1:=a1^2-s-(s-1)*a1;rel1; rel2:=a2^2-t-(t-1)*a2;rel2; rel3:=(a2*a1)^2-s*a6-(s-1)*a5-s*a3;rel3; rel4:=(a1*a2)^2-s*a6-(s-1)*a5-s*a4;rel4; def1:=a1*a2-A.4;def1; def2:=a2*a1-A.3;def2; def3:=a1*a2*a1-A.2;def3; def4:=(a2*a1*a2)-(a1*a2*a1)-A.1;def4; ID:=ideal; G:=GroebnerBasis(ID);G;#G; //Gr\"obner basis for the association scheme of a 2-(v,k,1) //design with parameters (t,s) a6^2 -(t^2s-2ts^2-2ts-t+s^3+3s^2+2s)*a6 -(t^2s-2ts^2-ts+s^3+s^2)*a5-(t^2s-2ts^2-ts+s^3+s^2)*a4 -(t^2s-2ts^2-ts+s^3+s^2)*a3-(t^2s-2ts^2-ts+s^3+s^2)*a2 -(t^2s-ts^2)*a1-(t^2s-ts^2)*a0, a6*a5-(ts^2-s^3-s^2)*a6-(ts^2-ts-s^3+s^2)*a5 -(ts^2-ts-s^3+s^2)*a4 -(ts^2-s^3)*a3-(ts^2-s^3)*a2, a6*a4-(ts-s^2-s)*a6-(ts-t-s^2+s)*a5-(ts-t-s^2+s)*a4 -(ts-s^2)*a3-(ts-s^2)*a2,

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a6*a3-(ts-s^2-s)*a6-(ts-s^2)*a5-(ts-s^2)*a4, a6*a2-(t-s-1)*a6-(t-s)*a5-(t-s)*a4, a6*a1-(s)*a6, a5*a6-(ts^2-s^3-s^2)*a6-(ts^2-ts-s^3+s^2)*a5-(ts^2-s^3)*a4 -(ts^2-ts-s^3+s^2)*a3 -(ts^2-s^3)*a2, a5^2 -(s^3-s^2)*a6-(ts+s^3-3s^2+2s-1)*a5-(s^3-2s^2+s)*a4 -(s^3-2s^2+s)*a3 -(s^3-s^2)*a2-(ts^2-ts)*a1-(ts^2)*a0, a5*a4-(s^2)*a6-(s^2-2s+1)*a5-(s^2-s)*a4-(s^2-s)*a3-(s^2)*a2, a5*a3-(s^2-s)*a6-(s^2-2s+1)*a5-(s^2-s)*a4-(ts- s)*a3-(ts)*a1, a5*a2-(s)*a6-(s-1)*a5-(s)*a4, a5*a1-(s-1)*a5-(s)*a3, a4*a6-(ts-s^2-s)*a6-(ts-s^2)*a5-(ts-s^2)*a3, a4*a5-(s^2-s)*a6-(s^2-2s+1)*a5-(ts-s)*a4-(s^2-s)*a3-(ts)*a1, a4^2 -(s)*a6-(s-1)*a5-(s)*a3, a4*a3-(s-1)*a6-(s-1)*a5-(ts-s)*a2-(ts)*a0, a4*a2-(1)*a6-(1)*a5, a4*a1-(s-1)*a4-(s)*a2, a3*a6-(ts-s^2-s)*a6-(ts-t-s^2+s)*a5-(ts-s^2)*a4-(ts-t-s^2+s) *a3-(ts-s^2)*a2, a3*a5-(s^2)*a6-(s^2-2s+1)*a5-(s^2-s)*a4-(s^2-s)*a3-(s^2)*a2, a3*a4-(t-1)*a5-(ts-t)*a1-(ts)*a0, a3^2 -(s)*a6-(s-1)*a5-(s)*a4, a3*a2-(t-1)*a3-(t)*a1, a3*a1-(1)*a5, a2*a6-(t-s-1)*a6-(t-s)*a5-(t-s)*a3, a2*a5-(s)*a6-(s-1)*a5-(s)*a3, a2*a4-(t-1)*a4-(t)*a1, a2*a3-(1)*a6-(1)*a5, a2^2 -(t-1)*a2-(t)*a0, a2*a1-(1)*a4, a1*a6-(s)*a6, a1*a5-(s-1)*a5-(s)*a4, a1*a4-(1)*a5, a1*a3-(s-1)*a3-(s)*a2, a1*a2-(1)*a3, a1^2 -(s-1)*a1-(s)*a0 //fusion partitions for 2-(v,k,1) designs (with t>s), //except for those partitions with only t=1=s partition1=[[1,2,3,4,5,6]] [] %partition2=[[1,2,3,4,5],[6]] %[T-S] partition3=[[1,2,3,4,6],[5]]

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[S-1] %partition4=[[1,2,3,4],[5,6]] %[T-1] %partition6=[[1,2,5],[3,4],[6]] %[T-S,S^2-5*S+4]=[T-1,(S-4)(S-1)] %partition7=[[1,2],[3,4],[5],[6]] %[T-S] partition9=[[1,3,4,5,6],[2]] [] %partition11=[[1,3,4,5],[2],[6]] %[T-S] %partition12= [[1,3,4],[2,5],[6]] %[T-2,S-2] partition13=[[1,5],[2,3,4,6]] [S-2] %partition14=[[1,5],[2,3,4],[6]] %[T-2,S-2] partition15=[[1,6],[2,3,4,5]] [T-S-1] partition16=[[1],[2,3,4,5,6]] [] partition17=[[1],[2,3,4,5],[6]] [] partition18=[[1,6],[2,3,4],[5]] [T-2,S-1] partition20=[[1,6],[2,5],[3,4]] [T-4,S-3] partition21=[[1],[2,5],[3,4],[6]] [S-1] partition22=[[1,6],[2],[3,4,5]] [T-S-1] partition23= [[1],[2],[3],[4],[5],[6]] []

References 1. Z. Arad and M. E. Muzychuk (eds.) Standard Integral Table Algebras Generated by a Non-real Element of Small Degree. Lecture Notes in Math., Vol. 1773, Springer, Berlin, 2002. 2. B. Buchberger, Gr¨ obner bases, an algorithmic method in polynomial ideal theory, in N. K. Bose (ed.) Multidimensional System Theory, pp. 184–232, Reidel, Dordrecht, 1985.

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3. D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, Berlin, 2007. 4. I. A. Faradˇzev, M. H. Klin, and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, in Investigations in Algebraic Theory of Combinatorial Objects. Math. Appl. (Soviet Ser.), Vol. 84, pp. 1–152, Kluwer Academic, Dordrecht, 1994. 5. R. Kilmoyer and L. Solomon, On the theorem of Feit–Higman, J. Comb. Theory, Ser. A, 15 (1973), 310–322. 6. M. Klin, C. Pech, and P.-H. Zieschang, Flag Algebras of Block Designs, I. Initial Notions. Steiner 2-designs and Generalized Quadrangles, Math-AL10-1998, Technische Universit¨ at, Dresden, November 1998 7. R. A. Liebler, Tactical conﬁgurations and their generic ring, Eur. J. Comb., 9 (1988), 581–592. 8. The MAGMA computational algebra system for algebra, number theory and geometry, The University of Sydney Computational Algebra Group, http://magma.maths.usyd.edu.au/magma. 9. E. Martinez-Moro, Computations on character tables of association schemes, in Computer Algebra in Scientiﬁc Computing, CASC’99, Munich, pp. 293–307, Springer, Berlin, 1999. 10. U. Ott, Some remarks on representation theory in ﬁnite geometries, in Geometries and Groups. Lecture Notes in Math., Vol. 893, pp. 68–110, Springer, Berlin, 1981. 11. K. W. Smith, Flag algebras of a symmetric design, J. Comb. Theory, Ser. A, 48 (1988), 209–228. 12. J. Tits, Sur la trialit´e et certains groupes qui s’en d´eduisent, Inst. Hautes Etudes Sci. Publ. Math., 2 (1959), 14–60. 13. P. -H. Zieschang, On dihedral conﬁgurations and their Coxeter geometries, Eur. J. Comb., 18 (1997), 341–354.

Enumerating Set Orbits Christian Pech1 and Sven Reichard2 1

2

Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel. [email protected] University of Western Australia, Crawley 6009, Western Australia. [email protected]

Summary. We describe a practically eﬃcient canonicity test for orbit representatives of permutation group acting on sets. This allows us to perform a wide range of combinatorial searches. We describe an implementation of the algorithm in GAP. We give a few applications, one of which answers a question by De Wispelaere regarding the classiﬁcation of all two-ovoids in the classical generalized hexagon of order 4.

Key words: Set-orbits, Orderly generation, Computer algebra package, Spreads, Incidence structures, Fusions of coherent conﬁguration, Steiner triple systems, Generalized quadrangles, Generalized hexagons, Schur rings

1 Introduction Many combinatorial problems can be stated in the framework of subset problems: Given a ﬁnite set Ω ﬁnd all subsets S ⊆ Ω satisfying a given property. We will call these subsets solutions. Some examples: 1. Independent sets of a graph Γ : Here, Ω is the set of all vertices of Γ . Solutions are subsets S of Ω such that no two elements of S are adjacent. 2. Proper graph colorings: Here, Ω consists of all independent sets. Solutions are collections of sets which partition the set of vertices. 3. Arcs in projective planes: Ω is the set of points; an arc is a set of points such that no three are collinear. Often the problem exhibits some degree of symmetry. This can be described as a group G acting on Ω leaving the set of solutions of the problem invariant. In the examples above, the automorphism groups of the given

C. Pech was supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics at the Mathematics Department of Ben-Gurion University. S. Reichard was funded by ARC Discovery Projects Grant No. 35400300.

M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 4,

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graphs or projective planes serve as such groups of symmetries. The action of G on Ω induces an action of G on the solutions. Usually, one is interested in non-isomorphic solutions, i.e., in solutions from diﬀerent orbits under this action. This raises the question of how to enumerate orbits of G acting on subsets of Ω. The most straightforward method is to store all subsets of Ω, and compute all orbits as sets of subsets. This can be improved a bit by noting that all sets in the same orbit have the same number of elements; thus we can generate the orbits of sets of each size independently. This approach is used in Computer Algebra Systems like GAP [9]. The drawback is the memory requirement, which is exponential in the size of Ω, restricting |Ω| to currently around 30. Another approach is to search for representatives of set orbits incrementally, adding one element at a time. Here we note that once we encounter two equivalent sets S1 , S2 we may immediately discard one of them, since any solution containing S2 will be equivalent to a solution containing S1 . A depth-ﬁrst approach was ﬁrst described in by Zaichenko [21], based on the general idea of “orderly generation” [13]. This has the advantage that only a single set has to be kept in memory at any given time, making it very memory eﬃcient, at the cost of some computational eﬃciency. It relies on the concept of canonical orbit representatives and their algorithmical recognition. A general theoretical description of similar algorithms was given by Laue [11]; a breadth-ﬁrst implementation was used to construct t-designs by Schmalz [16]. In this article we describe another depth-ﬁrst approach which uses techniques of dynamic programming for an eﬃcient implementation. In Sect. 2 we give some combinatorial and algebraic preliminaries. In Sect. 3 we formulate the general problem. In Sect. 4 we give a straightforward implementation of a canonicity test; improvements that make it eﬃcient in practice are described in Sect. 5. In Sect. 6 we describe how to use this algorithm to enumerate set orbits. In Sect. 7 we give some details of an implementation in GAP. Finally, in Sect. 8, we give some applications to combinatorial problems.

2 Preliminaries 2.1 Finite Permutation Groups Let Ω be a ﬁnite set. A bijection g : Ω → Ω is called a permutation of Ω. We use an exponential notation for permutations, so for x ∈ Ω and a permutation g we denote the image of x under g by xg . The composition of two permutations g, h of Ω is again a permutation, denoted by gh. We follow the convention of writing compositions left-to-right, such that for x ∈ Ω, xgh = (xg )h .

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Given a permutation g its inverse mapping g −1 is again a permutation. For any g we have that gg −1 = g −1 g = e, where e denotes the identity mapping. The set of all permutations of Ω forms a group under these operations, the symmetric group S(Ω). Any subgroup G of S(Ω) is a permutation group acting on Ω. We call the elements of Ω points, and the number of points the degree of G. Given a set {g1 , . . . , gk } of permutations in S(Ω) we may consider the group generated by the permutations, i.e., the smallest subgroup of S(Ω) containing all of the given permutations. We denote this group by g1 , . . . , gk . Two points x, y ∈ Ω are said to be equivalent under a permutation group G if there is an element g ∈ G which maps x to y. Since G is a group this is in fact an equivalence relation. The equivalence classes are the orbits of G; we denote the orbit containing x as xG . Given a point x ∈ Ω, let Gx be the set of all elements g of G mapping x to itself. Gx is a subgroup of G; we call it the stabilizer of x under G. Similarly, given several points x1 , . . . , xk , we deﬁne the pointwise stabilizer Gx1 ,...,xk to consist of those elements ﬁxing each xi . Given a set S = {x1 , . . . , xk } ⊆ Ω, we denote by GS the set of all elements of G mapping S to itself, thus xgi ∈ S for all i. GS is a group, the setwise stabilizer or simply the stabilizer of S. It contains the pointwise stabilizer as a subgroup. A sequence x1 , . . . , xk is a base of G if its pointwise stabilizer is trivial. Given a base we deﬁne a chain of subgroups G(i) , i = 0 . . . , k, by G(0) = G, (i−1) G(i) = Gxi , such that G(i) is the pointwise stabilizer of the ﬁrst i points. We call this the stabilizer chain for the given base. A set of generators of G which also contains generators for all the G(i) is known as a strong generating set (relative to the given base). Stabilizer chains can be computed and are a basic tool for computations in permutation groups. For details, see [17, 2]. 2.2 Combinatorial Search Many problems in Combinatorics can be formulated in the following manner: Given a set of basic “components”, construct from them larger structures which satisfy certain properties. In many cases the larger structure is simply formed by taking an appropriate subset of the given components, however at times the construction is more involved. As an example, consider the construction of a projective plane of a given order n. Here, we are given a point set P , and we have to ﬁnd a set B of lines, i.e., point sets of size n + 1, such that each pair of distinct points appears together in exactly one line. Here, we can consider all subsets of size n + 1 as given (further on called “lines”), and we need to select several of those that will fulﬁll the axioms of projective planes. The naive way would be to look at all possible subsets and check whether the wanted properties are satisﬁed. However, this is feasible only in the small-

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est cases since the number of subsets grows exponentially with the number of points. The standard way to slow down this “combinatorial explosion” has been described under various names such as “backtrack search” and “branch and bound”. The idea is to construct solutions incrementally, checking after each step whether it is possible in theory to extend the current, partial solution to a complete solution. In the example of the projective plane, assume that in the beginning we pick two lines that intersect in two points. It is clear that no solution can contain these two lines simultaneously, so we may immediately discard this partial solution and search elsewhere. More formally, we start with a search space (a set) of candidate solutions on which is deﬁned a predicate SOL such that SOL(x) is true if and only if x is a solution. Moreover, we deﬁne a predicate PSOL for which we require that for y ⊆ x, SOL(x) implies PSOL(y). Thus if we ﬁnd that for a partial solution y, PSOL(y) does not hold we may discard all candidate solutions containing y. Candidate solutions that fulﬁll PSOL are called partial solutions. Returning to the projective planes we may require that for a partial solution y any two lines intersect in a single point. This will reduce the search space signiﬁcantly. It is always possible to ﬁnd such a partial predicate PSOL, by setting PSOL(y) to true for all y. In practice there is a trade-oﬀ between the cost of computing PSOL and its power to discard large parts of the search space.

3 Search and Symmetry Another important way to speed up a combinatorial search is by using symmetry. If the original conﬁguration of given components is symmetrical this means that the set of solutions reﬂects that symmetry. Usually one is interested only in solutions that are not equivalent under this symmetry. More formally, suppose that there is a group G acting on the given set which leaves the predicate SOL invariant, thus it maps solutions to solutions. We would like to ﬁnd inequivalent solutions, i.e., representatives of the Gorbits acting on all solutions. If we have an isomorphism test for solutions, i.e., a test whether two solutions are equivalent, then we can construct all solutions and discard those that are isomorphic to those found earlier. This approach has two drawbacks. Firstly we need to store all solutions (or at least all representatives). Secondly we still need to search equivalent portions of the search tree multiple times. We can get around the second problem by not only storing the full solutions, but also the partial solutions, and compare each partial solution to everything previously encountered. If we ﬁnd that the current partial solution is already known, then we can again discard the remaining part of the search tree. However, this requires even more storage, and a quadratically growing number of isomorphism tests.

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A way around both problems is the use of canonical representatives. Suppose we have another predicate CSOL that is true for exactly one element of each orbit of solutions; we call this particular element canonical. Then we may reject a solution if it is not canonical, without the need to compare it to other solutions. Extending this idea to partial solutions requires some care in order not to lose any solution. In particular we need to make sure that for each canonical solution c there is a chain of canonical partial solutions leading up to c. If we can achieve this then we only have to consider canonical partial solutions, and we will obtain all canonical solutions. This approach is known as orderly generation, as described by Read [13] and Faradˇzev [6]. Canonicity tests have been developed for several combinatorial structures, such as graphs [12] and designs [10]. Here, we will describe such a test for subsets in general. This will allow to perform orderly generation for any problem that may be formulated in terms of subsets.

4 The Basic Algorithm Let G be a permutation group acting on a set Ω, and let T ⊂ Ω be a set of points. Assume that we have a linear order ‘<’ on Ω and as a consequence a natural lexicographical order on the subsets of Ω. We say that T is canonical (under G) if it is lexicographically smallest in its orbit (by deﬁnition, the empty set is canonical in its orbit because it is the only element of its orbit). The following lemma shows that this concept of canonicity is well-behaved: Lemma 1. Let G be group acting on Ω, ∅ = T ⊂ Ω, t = max T , and T = T − t. If T is canonical, then so is T . Proof. By way of contradiction assume that T is not canonical. Then there is an element g ∈ G such that S = T g is lexicographically smaller that T . Let S = T g , then S = S ∪ {tg }. This is lexicographically smaller than T , contradicting the canonicity of T . Thus the problem to be solved is to check whether a given set is lexicographically smallest in its orbit. This problem has been previously studied in [1]. Instead of solving the original question whether under the action of a permutation group G a given set T is mapped to any smaller set we look at the slightly more general problem of mapping a set S to any set smaller than another given set T . If S = T this solves the original question; however in the implementation we use the more general approach recursively. Now a set S is mapped to a set smaller than T if either some element of S is mapped to an element smaller than anything in T , or if some element of S is mapped to the minimal element of T , and the remainder of S is mapped to

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Algorithm 1: IsReducibleSet1(G, T ) return IsMappedToSmallerSet1(G, T, T )

Algorithm 2: IsMappedToSmallerSet1(G, S, T ) Check if any permutation in G maps the set S to a set which is lexicographically smaller than the set T Data: A permutation group G, two sets S and T Result: A boolean value if T is empty then return false; if S is empty then return true; t0 ← min T ; if ∃g ∈ G, ∃s ∈ S : sg < t0 then return true; for s ∈ S do if ∃g ∈ G : sg = t0 then S ← (S \ {s})g ; T ← T \ {t0 }; G ← Gt0 ; // point stabilizer if IsMappedToSmallerSet1(G , S , T ) then return true return false;

something smaller than the remainder of T . This leads to the straightforward Algorithm 2. The double existence quantiﬁers in the middle of the algorithm amount to computing the orbits under G of the elements of S. Similarly, the ﬁnal loop has to cover only those elements of S which lie in the orbit of t0 under G.

5 Improvements 5.1 Recycling Information If we look at Algorithm 2, and in particular at its parameter T , we note except for removing the ﬁrst element between one level and the next, this set is never changed. This allows us to precompute a lot of the data needed in the algorithm. First, we can compute the various stabilizers occurring in the algorithm. If we assume that T = {t1 , . . . , tk }, then let G(i) denote the stabilizer of the ﬁrst i points in G (in other words, the G(i) form a stabilizer chain). Together with the stabilizers we can compute Schreier vectors for the various levels. (Schreier vectors are data structures that eﬃciently encode the

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coset representatives of a point stabilizer in the complete group. For a detailed description, see [4].) This allows us to eﬃciently compute an element of G(i−1) mapping ti to a given point. Finally, the fact that G(i−1) maps a given point x to something smaller than ti also depends only on i and x. Thus, we may either save this information as it comes up, or we can precompute it before starting the recursion. 5.2 Computing the Stabilizer In the algorithms above the termination condition is that the set T is empty. In fact, since in each invocation the sizes of S and T are equal, this also means that S is empty. This means that we have mapped the points of the initial S one by one to the points of T . In the case that we invoked the top level with S = T this means that we have found an element in the setwise stabilizer of T under G. Using a common ﬁrst-in-orbit argument we can break oﬀ the search at that point, tracking back to the ﬁrst point of T that was not moved. This allows us to reduce the search space by using symmetry. In fact, we may use the permutations found in that way, and construct a strong generating set of the setwise stabilizer of T under G step by step. This is given as Algorithm 3 and Algorithm 4.

Algorithm 3: IsReducibleSet2(G, T ) Check whether T is canonical under G; compute the stabilizer H of T in G in the process. Note that if T is reducible, H will be a possibly proper subgroup of the stabilizer. Let H be the trivial group; for i = |T |, |T | − 1, . . . , 1 do Let G be the stabilizer of the ﬁrst i − 1 elements of T ; Let T be T with the ﬁrst i − 1 elements removed; Let e be the trivial permutation; if IsMappedToSmallerSet2(G , T , T , e) then return true; Now, GT = H; return false;

5.3 Using the Stabilizer Assume that during the search we know some group H that ﬁxes the set S, hence a subgroup of the setwise stabilizer GS . We select an element s ∈ S, map it to t0 , and ﬁnd that this doesn’t lead to S being mapped to something smaller than T . If s is in the orbit sH , then there is an element h ∈ H which maps s to s . Then, mapping s to t will not lead to a reduction neither.

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Algorithm 4: IsMappedToSmallerSet2(G, S, T, g0 ) Check if any permutation in G maps the set S to a set which is lexicographically smaller than the set T ; compute the stabilizer of the original set in the process. Data: A permutation group G, two sets S and T , a permutation g0 Result: A boolean value if T is empty then Add g0 as a new generator to the known stabilizer H; return to Algorithm 3; if S is empty then return true; t0 ← min T ; if ∃g ∈ G, ∃s ∈ S : sg < t0 then return true; for s ∈ S do if ∃g ∈ G : sg = t0 then S ← (S \ {s})g ; T ← T \ {t0 }; G ← Gt0 ; // point stabilizer if IsMappedToSmallerSet2(G , S , T , g0 · g) then return true return false;

Thus, we use the information about the stabilizer of T in G as described in the previous section to compute part of the stabilizer of S as follows: Removing a point from S amounts to ﬁnding the stabilizer of that point; moving S under a permutation g requires conjugating the stabilizer by that permutation. This method gives us the required subgroup of GS to work with. This allows to prune the search tree at the expense of conjugating stabilizer chains, and ﬁnding point stabilizers. It turns out that conjugation is quite expensive, and it can actually be avoided. 5.4 Avoiding Conjugation Above we described how the stabilizer GS of the set S is used to prune the search tree. When S is mapped under a permutation g, the stabilizer GS needs to be conjugated. This involves the conjugation of all generators and hence it is computationally expensive. However, the stabilizer is only used to test whether two elements of S are equivalent. If for each element of S we keep track of its origin, i.e., of its preimage in the original set, then we can check the equivalence of the preimages by using a stabilizer of the original group, without the need of any conjugation. This gives the ﬁnal version of the algorithm, Algorithm 5.

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Algorithm 5: IsMappedToSmallerSet3(G, S, S0 , T, g0 , H) Check if any permutation in G maps the set S to a set which is lexicographically smaller than the set T ; compute the stabilizer of the original set in the process. Data: A permutation group G, sets S, S0 , and T , a permutation g0 , and a subgroup H of G (a subgroup of the stabilizer of the original set) Result: A boolean value if T is empty then Add g0 as a new generator to the known stabilizer H; return to Algorithm 3; if S is empty then return true; t0 ← min T ; if ∃g ∈ G, ∃s ∈ S : sg < t0 then return true; for i ∈ 1, . . . , |S| do Let s be the i-th element of S; Let s0 be the i-th element of S0 ; if s0 is marked as used then continue; if ∃g ∈ G : sg = t0 then S ← (S \ {s})g ; S0 ← S0 \ {s0 }; T ← T \ {t0 }; G ← Gt0 // point stabilizer H ← Hs0 ; if IsMappedToSmallerSet3(G , S , S0 , T , g0 · g, H ) then return true Add all new generators of H to H; Mark all elements of sH 0 as used; return false;

6 Enumerating Set Orbits Given the algorithm described above, we can now enumerate the orbits of (G, Ω) acting on the power set 2Ω , by using Lemma 1 and the following: Lemma 2. If S is canonical, then max S is minimal in its orbit under GS . Proof. Suppose that x = max S is not minimal. Then there is a g ∈ GS such that y = xg < x. But then S g = (S ∪ {x})g = S g ∪ {xg } = S ∪ {y} < S, and since g ∈ G, this contradicts the canonicity of S. We get the following algorithm (see Algorithm 6). There are a couple of things to note here: • The stabilizer H needed here is obtained as a byproduct in the canonicity test.

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Algorithm 6: EnumerateSubsets(G, Ω, S = ∅) H ← GS ; orbitReps ← {x ∈ Ω : max S < x ≤ xh ∀h ∈ H}; for x ∈ orbitReps do S˜ ← S ∪ {x}; ˜ then if not IsReducibleSet(G, S) ˜ output S; ˜ EnumerateSubsets(G, Ω, S);

• Much of the precomputed data used in the canonicity test for S, such as the stabilizer chain and the reducible orbits for all but the ﬁnal level, can ˜ which is obtained by adding a single be reused in the test for the set S, element to S. • The point stabilizer Hx is a large subgroup of the stabilizer GS˜ ; in fact it is the stabilizer of x in that group. So, injecting Hx in the test for S˜ will speed up that procedure.

7 Implementation The algorithm as described above has been implemented and tested in GAP [9]. The user has access to it on three diﬀerent levels, providing a trade-oﬀ of control and ease of use. The most straightforward access to the algorithm is a function IsCanonicalSetOrbitRepresentative, which takes as arguments a permutation group G and a set T , and returns a boolean value indicating whether T is smallest in its orbit. gap> G := Group((1,2,3,4,5), (2,5)(3,4)); Group([ (1,2,3,4,5), (2,5)(3,4) ]) gap> IsCanonicalSetOrbitRepresentative(G, [1,2]); true gap> IsCanonicalSetOrbitRepresentative(G, [1,3]); true gap> IsCanonicalSetOrbitRepresentative(G, [1,5]); false Any additional information obtained during the execution of the algorithm, such as the stabilizer of T , is lost. If this information is needed, a more involved invocation is needed: can := CanonicityTest(G, [1,2], 5);; gap> result := RunTest(can); true

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gap> stabilizer := GetStabilizer( can ); Group( [ (1,2)(3,5) ] ) Finally, for the common case that some action is to be performed for each representative it is possible to deﬁne an object representing the domain of all such representatives. For this we need to provide the group G, the set on which G acts, and the desired sizes of the sets to consider. Hence for example we can ﬁnd all orbits of sets of size 2 or 3 by the following: gap> domain := [1..5]; [ 1 .. 5 ] gap> sizes := [2,3]; [ 2, 3 ] gap> representatives := SetOrbitRepresentatives(G, domain, sizes);; gap> Size(representatives); 4 gap> for set in representatives do > Print( set, "\n"); > od; [ 1, 2 ] [ 1, 2, 3 ] [ 1, 2, 4 ] [ 1, 3 ]

8 Applications We tested the algorithm mainly with two types of problems: Enumeration of spreads in incidence structures, and enumeration of fusions in coherent conﬁgurations. An incidence structure can be thought of as a generalized geometry. It consists of two types of objects, “points” and “lines”, and some incidence relation between the two types. In other words, each point is either incident to a given line, or it is not incident to it. The notions of point and line suggest using language from planar geometry, thus, if a point P and a line L are incident, we say that P lies on L, and that L goes through P . Given an incidence structure, its dual is obtained by interchanging the roles of points and lines, and reverting the incidence relation accordingly. Thus, a point of the original becomes a line of the dual, and vice versa. A spread in an incidence structure is a subset of lines which covers each point exactly once. In particular, any two lines in a given spread are disjoint. The horizontal lines form a spread of the euclidean plane R2 .

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8.1 STS(15) For the spreads we ran several test cases with known results. One of them was the classiﬁcation of Steiner triple systems on 15 points. It is a well known result [20] that there are exactly 80 such structures up to isomorphism. This can be translated to a spread problem as follows: Take as points all unordered pairs of a 15-element set; as lines take triples. Deﬁne incidence “points” and 15 as inclusion. This gives an incidence structure with 15 2 3 “lines”. A Steiner triple system on the original set is a set of triples such that each pair of elements is contained in exactly one triple. In other words, each “point” is incident to exactly one “line”, thus we have a spread. As a group we have the symmetric group of degree 15 acting on the original elements. The algorithm had to deal with the group acting on triples, which gives a degree of 455. The construction of all non-isomorphic solutions took 6 minutes of CPU time.3 8.2 GQ(3, 9) We also determined all spreads in the unique generalized quadrangle GQ(3, 9). We found a total of 26 orbits of solutions, which conﬁrms a result by A. E. Brouwer [3]. The computation took 2 s.. The group in question has degree 112. These spreads are interesting also from another point of view. The point graph of the generalized quadrangle is the ﬁrst subconstituent of the McLaughlin graph of order 275, which corresponds to a rank 3 representation of the McL simple group. It is still an open question whether this bigger graph is the point graph of a suitable partial geometry pg(5, 28, 2) (see [19, 15, 18]). Supposing that such a geometry exists, we call a point P together with all lines through it a bundle. If we take a bundle and remove the point P we obtain a set of lines which covers each neighbor of P exactly once. In other words, we get a spread in the ﬁrst subconstituent, which we know is isomorphic to GQ(3, 9). However, the group acting here is the stabilizer of a point in M cL, which is smaller than the full automorphism group of the generalized quadrangle. In fact, some of the orbits of spreads split, and we get a total of 36 orbits of possible bundles. 8.3 Generalized Hexagon In her thesis, A. De Wispelaere [5, p. 159] noted that there are two known two-ovoids in the classical generalized hexagon of order 4. The question was raised whether there are any others. Two-ovoids can be regarded as a spread of the dual structure, hence they ﬁt in this framework. The number of points in the generalized hexagon is 3

The computation was performed on an Intel Celeron processor at 1.6 GHz.

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−1 1365 = 44−1 , same as the number of lines. Each line contains 5 points, and each point lies on 5 lines. A two-ovoid is a set of points meeting each line exactly once; thus it contains 1365/5 = 273 points. In the dual hexagon this corresponds to a set of lines containing each point exactly once, i.e., a spread. We were able to give a negative answer to this question; all two-ovoids are known. The search took approximately 2 weeks of CPU time. 6

8.4 Schur Rings Over Small Groups Schur rings or S-rings are structures related to (abstract) groups (see, e.g., [7, p. 56]). They form a special case of association schemes which are central to the area of Algebraic Combinatorics. The construction of S-rings over a given group G relies on a search of certain subsets of G. Hence, here we consider the action of G on itself. In [8], all S-rings of order up to 31 were determined, as well as those of order 33 through 35. The S-rings of order 32 were computed by the ﬁrst author using an earlier version of the above described algorithm. Recently the second author, using the algorithm described above, was able to extend this to all orders up to 47. Details will follow elsewhere [14].

9 Conclusion We have described a fast canonicity test for sets under the action of permutation groups. This test will be made available as a GAP share package. It allows to treat a wide range of combinatorial problems in a uniform manner. The authors wish to thank M. Klin, A. Niemeyer, and C. Praeger for helpful input. The authors are pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria for partial support of this project. The work of the second author has been funded by the Australian Research Council. Last but not least, the authors would like thank the anonymous referees for their helpful remarks and suggestions.

References 1. L. Babai and E. M. Luks, Canonical labeling of graphs, in Proc. 15th ACM STOC, pp. 171–183, 1983. 2. L. Babai, E. M. Luks, and A. Seress, Fast management of permutation groups, in 29th Annual Symposium on the Foundations of Computer Science, pp. 272–282, IEEE Press, New York, 1988.

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3. A. E. Brouwer, Homepage, http://www.win.tue.nl/∼aeb. 4. G. Butler, Fundamental Algorithms for Permutation Groups, Lect. Notes Comp. Sci., Vol. 559, Springer, Berlin/Heidelberg, 1991. 5. A. De Wispelaere, Ovoids and Spreads of Finite Classical Generalized Hexagons and Applications, Ph.D. Thesis, Universiteit Gent, 2006. 6. I. A. Faradˇzev, Constructive enumeration of combinatorial objects, in Probl`emes combinatoires et th´eorie des graphes, Univ. Orsay, Orsay, 1976. Colloq. Internat. CNRS, Vol. 260, pp. 131–135, CNRS, Paris, 1978. 7. I. A. Faradˇzev, M. H. Klin, and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, in I. A. Faradˇzev, et al. (eds.) Investigations in Algebraic Theory of Combinatorial Objects. Math. Appl. (Soviet Ser.), Vol. 84, pp. 1–152, Kluwer Academic, Dordrecht, 1994. 8. F. Fiedler, Enumeration of Cellular Algebras Applied to Graphs with Prescribed Symmetry, Master’s thesis, Technische Universit¨ at Dresden, 1998. 9. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10 ; 2007, http://www.gap-system.org. 10. A. V. Ivanov and I. A. Faradˇzev, Constructive enumeration of incidence systems. I, II, III, Rostock. Math. Kolloq., 24 (1983), 4–62 (in Russian). 11. R. Laue, Construction of combinatorial objects – a tutorial. Bayreuther Math. Schr., 43 (1993), 53–96. 12. B. D. McKay, The nauty package, http://cs.anu.edu.au/people/bdm/ nauty/. 13. R. C. Read, Every one a winner, or how to avoid isomorphism search when cataloguing combinatorial conﬁgurations, Ann. Discrete Math., 2 (1978), 107–120. 14. S. Reichard, S-rings of order up to 47, in preparation. 15. S. Reichard, An Algorithm for the Construction of Partial Geometries with Given Point Graphs, Technical Report, Technische Universit¨ at Dresden, MATH-AL-12-1997, October 1997. 16. B. Schmalz, t-Designs zu vorgegebener Automorphismengruppe, Dissertation, Universit¨ at Bayreuth, 1992. 17. C. C. Sims, Computational methods in the study of permutation groups, in Computational Problems in Abstract Algebra, pp. 169–183, Pergamon, Oxford, 1970. 18. L. H. Soicher, Is there a McLaughlin geometry? J. Algebra, 300 (2006), 248–255. 19. J. H. van Lint, On ovals in P G(2, 4) and the McLaughlin graph, in Papers Dedicated to J.J. Seidel, pp. 234–255, Eindhoven, 1984. 20. H. S. White, F. N. Cole, and L. D. Cummings, Complete classiﬁcation of the triad systems on ﬁfteen elements, Mem. Nat. Acad. Sci. U.S.A. (2nd memoir), 14 (1919), 1–89. 21. V. A. Zaichenko, An Algorithmic Approach to the Construction of Combinatorial Objects and Computations in Permutation Groups Based on the Method of Invariant Relations, PhD thesis, Moscow, 1981 (in Russian).

The 2-dimensional Jacobian Conjecture: A Computational Approach Ronen Peretz Department of Mathematics, Ben-Gurion University, Beer Sheva, Israel. [email protected]

Summary. The Jacobian Conjecture is one of the most important open problems in algebraic geometry. It was included among the millennium open problems in mathematics by Steve Smale in his address to the Millennium ICM (in the year 2000). Given a polynomial mapping F : Cn → Cn which has a nonzero constant determinant of its Jacobian matrix (The Jacobian Condition), the conjecture is that F is an invertible morphism. This means that it is injective, surjective and that its inverse mapping F −1 is also polynomial. Even in dimension n = 2 this is still open. This article is about the 2-dimensional Jacobian Conjecture. The degree of a polynomial mapping F is the maximum degree of its polynomial coordinate functions. A striking fact that we prove is that given a degree d, the 2-dimensional Jacobian Conjecture can be settled for all the polynomial mappings of degree d or less. If it is disproved then a counterexample is constructed. The magic tool used is the machinery of Gr¨ obner bases. We use the powerful tools of computational algebra and in particular the theory of Gr¨ obner bases in order to solve a certain ideal membership problem in an algebra of many variables polynomials over the complex ﬁeld. The Jacobian condition satisﬁed by the mapping F (of degree d or less) is used in a canonical way to construct an ideal (the Jacobian ideal) within this algebra of polynomials. We consider the two relative resultants of the mapping F (one with respect to X and the second with respect to Y ). The key theorem we prove is that the Jacobian Conjecture is valid for F if and only if the leading coeﬃcients of these two resultants belong to the Jacobian ideal. We call this result: The resultant reformulation of the Jacobian Conjecture. Now the Gr¨ obner bases machinery comes in to help to decide on the ideal membership problem we have. The algorithm was programmed and was used to prove the 2-dimensional Jacobian Conjecture up to degree 15. The theoretical importance of this algorithm is that it shows that the conjecture is a decidable problem. The article contains many other important results, both new and old that are relevant to this particular computational approach to this famous open problem. We assume that the reader has some background in the theory and the folklore of the Jacobian Conjecture. The classical paper [H. Bass, E. Connell, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (2), 7 (1982), 287–330] and the only comprehensive book in this area [A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture,

M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 5,

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Progress in Mathematics, Vol. 190, Birkh¨ auser, Basel, 2000] are excellent sources. We use the notation introduced in those sources whenever possible.

´ Key words: The Jacobian Conjecture, Etale morphisms, Inversion formulas, Polynomial automorphisms, Asymptotic values

1 Introduction A polynomial map F : Cn → Cn is a map F = (P1 , . . . , Pn ) whose coordinate functions are polynomials over the complex ﬁeld C in the n variables (X1 , . . . , Xn ). We write (P1 , . . . , Pn ) ∈ C[X1 , . . . , Xn ]n , where C[X1 , . . . , Xn ] is the algebra of polynomials in (X1 , . . . , Xn ) over C. The polynomial map F is called invertible, or an automorphism of Cn if it is injective, surjective and its inverse mapping G = F −1 is also polynomial. This means that F ◦ G = G ◦ F = idCn where G is polynomial. Such an automorphism preserves the algebro-geometric structure of the important aﬃne variety Cn and hence its importance in algebraic geometry. It is natural to look for convenient criteria for a polynomial mapping to be invertible. Application of the chain rule to the functional equations F ◦ G = G ◦ F = idCn gives the following equations on the Jacobian matrices of F and G, JF (G)JG (X) = JG (F )JF (X) = In . Here we use the notation X = (X1 , . . . , Xn ). Taking determinants we get det JG (F ) · det JF (X) = 1. In particular the polynomial det JF (X1 , . . . , Xn ) cannot have a zero and the Fundamental Theorem of the Algebra implies that det JF (X) is a nonzero constant. We write det JF (X) ∈ C∗ and call this the Jacobian condition. So far we obtained the implication F is an automorphism

det JF (X) ∈ C∗ .

⇒

This naturally leads to consider the reverse implication det JF (X) ∈ C∗ and F polynomial

⇒

F is an automorphism.

In dimension n = 1 this is true and is easy to prove. For n ≥ 2 this is one of the famous open problems in mathematics, known as the Jacobian Conjecture. It appeared in the paper [11] by Ott-Heinrich Keller. An excellent survey of the problem appears in [2]. Also, a comprehensive book on the subject (as well as on related topics) is the one by Arno van den Essen [6]. A wealth of results were discovered over the years by many mathematicians that tried to resolve the Jacobian Conjecture. It is known to be true in dimension n = 2 up to about degree d = 100, [15]. Here the degree of a polynomial mapping is the maximum of the degrees of its coordinates. It is known to be true for quadratic polynomial mappings, i.e. d ≤ 2, in any dimension n, [2, 6]. Amazingly enough it was proved that if the Jacobian Conjecture is

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valid for polynomial mappings of degree d ≤ 3 in any dimension n, then it is valid for all degrees. The technique that was used to prove that result is called degree reduction. It is based on a K-theoretic principle, [2, 6]. This gives the impression that “we are almost there”. After all only maps of degree d = 3 should be considered (and even here, only very special subclass of polynomial mappings will suﬃce). In spite of this feeling the Jacobian Conjecture is still open. This is a peculiar property of that riddle in mathematics. It gives one the impression of a not so diﬃcult problem, and attracts mathematicians from various ﬁelds of research. Algebraic geometers as well as researchers in commutative algebra were involved. But also analysts that come from several complex variables, geometric function theory and geometers that do diﬀerential geometry tried to solve the problem. Topological methods were tried as well. Starting from manifold theory, varieties and complex spaces. Algebrotopological methods were used too. Representation theory and even theories on inﬁnite dimensional varieties were applied. The list is still not complete! But it turns out to be a diﬃcult problem. The by product of that, is the existence of many faulty “proofs” of this conjecture that appeared over the years. Some were even formally sent for publication and appeared in good mathematics journals. It is rather clear that the Jacobian Conjecture is a problem about polynomial mappings and over ﬁelds of characteristic zero. Thus even though it is tempting to think about a broader class of mappings (such as entire mappings on Cn ) the conjecture is known not to be true there. Thus there must be something very algebraic involved in the solution of this open problem. This might help us understand the origin of some faulty proofs. The author heard serious mathematicians claiming that “there is no obvious reason why such a conjecture should be true” and hence they believe that it is false. However, it might be that this is one of these stubborn problems that have a counterexample only in a very large dimension or degree, way beyond our computer technology, and hence it might be extremely diﬃcult to construct one. One needs to look at the counterexample of S. Pinchuck, [23], for the so called “Real Jacobian Conjecture” to have a feeling of what might be involved. The current paper describes an eﬀort to deal with the Jacobian Conjecture in dimension n = 2. It is a follow up of a previous attempt that was geometric in nature. This geometric approach is described in [20, 21] and in particular in [22]. We outline the geometric ideas that are involved in this former attempt. A polynomial map F that satisﬁes the Jacobian condition det JF (X) ∈ C∗ is called an ´etale mapping. From the point of view of diﬀerential geometry an ´etale mapping is a polynomial local diﬀeomorphism of Cn . Hence we can restate the Jacobian Conjecture in this new language as follows, F ´etale

⇒

F is an automorphism.

It is useful to recall that injective polynomial mappings Cn → Cn are surjective and their set theoretic inverse mapping is polynomial. So it is really only the injectivity property that we are after. The above restatement means that

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a local condition (being ´etale) implies for polynomial mappings over C the corresponding global property (of being a global diﬀeomorphism). A beautiful theory developed by J. Hadamard, [9], tells us that in order to prove that a local diﬀeomorphism is in fact a global one, it suﬃce to exclude the existence of asymptotic values (these are ﬁnite limits at inﬁnity) of the mapping. This is the starting point of the geometric approach mentioned above. When one specializes the general Hadamard’s theory to polynomial mappings, one immediately comes across fractional power series expansions of the asymptotic curves of the mapping. A very elegant set of results along these lines were discovered by Nguyen Van Chau. As opposed to this approach the methods of the current paper are very algebraic and in some cases are even algorithmic. Thus computer algebra methods such as the computation of Gr¨obner bases of ideals are naturally used here. The connection to the previous (geometric) approach is via the study of asymptotic values of polynomial maps as suggested by J. Hadamard. Section 2 elaborates this algebraic version of Hadamard general theory. We restrict our attention to the 2-dimensional case of polynomial mappings F = (P (X, Y ), Q(X, Y )) : C2 → C2 . We translate the geometric study of asymptotic values into an algebraic study of the pair of resultants that are induced by F . If α, β are two new indeterminates then we can form the following two (so called) relative resultants, R(X) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = RN (α, β)X N + · · · + R0 (α, β), S(Y ) = resultant(P (X, Y ) − α, Q(X, Y ) − β, X) = SM (α, β)Y M + · · · + S0 (α, β). A main result that we prove in Sect. 5 (Theorem 5) is called The Resultant Formulation of the Jacobian Conjecture. It shows that (under some standard normalization on F ) the Jacobian conjecture is true iﬀ the Jacobian condition on F implies that the coeﬃcients RN (α, β), SM (α, β) of the above resultants are nonzero constants. For related results see [6]. This suggests that there should be some kind of algebraic relation between the Jacobian of F , ∂(P, Q)/∂(X, Y ) and the above resultants R(X), S(Y ) (provided that the conjecture is true!). In spite of many eﬀorts the author was unable to ﬁnd such a relation. However, the above method was fruitful and led to theorems that are reported in this paper. We now proceed to describe in some details the results that appear in the various sections of this paper. In Sect. 3 it is proved that if F : C2 → C2 is polynomial then any sequential asymptotic value of F is also an asymptotic value of F (Theorem 1). The proof relies on the technique of maximal domains [18, 19]. In Sect. 4 we prove some relations between the asymptotic values of a polynomial F : C2 → C2 and the zeros of the highest coeﬃcients of the above pair of resultants. In fact we show that the set of all the asymptotic values

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of F is the union of the two algebraic curves RN (α, β) = 0, SM (α, β) = 0 (Theorem 2). We also deduce that the set of X-inﬁnite asymptotic values of F form the curve RN (α, β) = 0 and the set of Y -inﬁnite asymptotic values of F is SM (α, β) = 0 (Theorem 3). This should be compared with the results in [20] for the real case. Thus we see that the existence of asymptotic values of a polynomial map F = (P, Q) is controlled by the structure of the highest coeﬃcients of the induced pair of resultants. Once again we mention the related results of Nguyen Van Chau [7, 25]. If F satisﬁes the Jacobian condition ∂(P, Q)/∂(X, Y ) ∈ C∗ then according to Sect. 2 F is invertible provided that it has no asymptotic values. Hence there should be a reformulation of the Jacobian conjecture in terms of the induced resultants. Indeed that Resultant Reformulation is proved in Sect. 5, Theorem 5: ∂(P, Q)/∂(X, Y ) ∈ C∗

⇔

RN (α, β), SM (α, β) ∈ C∗ .

Probably the core of the Jacobian conjecture (if true) is to explore the algebraic relation that should exist between the Jacobian of a map and it’s induced resultants. As a ﬁrst application of the Resultant Reformulation we prove in Sect. 6, Theorem 6 that the Jacobian conjecture is decidable. Namely, we describe an algorithm which gets as an input a positive integer d and decides (after ﬁnitely many steps) if the Jacobian conjecture holds true for all polynomial maps C2 → C2 of degree at most d. Moreover, if the answer is negative, then the algorithm generates a counterexample. The algorithm utilizes Buchberger’s algorithm for calculating a Gr¨ obner basis for the Jacobian variety of degree d [3, 8]. Naturally the complexity of our algorithm is high because the Buchberger-type algorithms are of high complexity [13]. The algorithm was implemented by the author (while visiting the University of Michigan) on a Sun platform. He used the standard Gr¨ obner package in Maple. The input to the program was a positive integer d and the output was a true/false ﬂag. If true then the Jacobian conjecture was found to be valid for all the complex polynomial mappings in dimension two and of degree d or less. A false ﬂag meant the other alternative and a counterexample of degree d or less would have been given. The package was able to automatically prove the conjecture for mappings of degree 15 or less. The running time was about 40 h. The Resultant Reformulation suggests natural inductive approach to prove the Jacobian conjecture. In Sect. 7 we discuss several such possibilities but these fail. This might indicate that the Jacobian conjecture is not true. In Sect. 8 we explore some covering properties of a polynomial F : C2 → C2 in terms of it’s induced resultants. For example in Theorem 7 we prove what might be called “a polynomial Picard’s Little Theorem”, namely, that there exist ﬁnitely many polynomials

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RN (α, β), . . . , R1 (α, β) (the highest N coeﬃcients of R(X)) such that C2 − F (C2 ) = VC (RN , . . . , R1 ). So generically F (C2 ) is the whole of C2 except, maybe, for a ﬁnite set whose size is controlled by the Bezout bound. Motivated by the Resultant Reformulation we pose the following conjecture: ∂(P, Q)/∂(X, Y ) has a zero on F −1 (RN (α, β) = 0). This conjecture implies the Jacobian conjecture. We show that a slightly stronger version of this conjecture fails (Example 5) which shows how delicate the situation is. For some results of Sect. 7 we use properties of certain well known derivations that are related to the Jacobian of a polynomial map. For that purpose a small theory on grading an algebra with a derivation is developed in Sects. 9, 10, 11 and 12. See related results in the book [6]. If k is a ﬁeld of characteristic 0 and A is a k-algebra with 1 and D a derivation of A, then there is a natural gradation of A with respect to D: the elements of class n in the gradation are those that are annihilated by n iterations of D. If we assume, further, the existence of an element q ∈ A for which D(q) = 1 (called a slice) then in “reasonable” algebras the multiplication of an element in class n by q is an element of class n + 1 and conversely (Theorem 8). As a consequence we obtain a structure theorem for the classes of this D-gradation in terms of ker(D, A) and of q (Theorem 9). This leads immediately to the well known fact that Nil (D) = ker(D, A)[q] which was noted by several people [23, 25] and [17] (Corollary 1). It also gives a necessary and suﬃcient condition for D to be a locally nilpotent derivation of A (Corollary 6). One application of the above is for the polynomial ring over k in n variables, A = k[x1 , . . . , xn ]. Given n elements f1 , . . . , fn ∈ A that satisfy the Jacobian condition, namely, that ∂(f1 , . . . , fn )/∂(x1 , . . . , xn ) = 1 one can take D = ∂(f1 , . . . , fn−1 , ·)/∂(x1 , . . . , xn ), q = fn and apply the above. This is closely related to the results on the derivations d1 , . . . , dn that were introduced in [17], however, the methods of proofs are diﬀerent. Section 13 is dedicated to invertible morphisms F : C2 → C2 and the structure of their resultants. The main result is that resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = R1 X + R0 (α, β) where R1 ∈ C∗ (Theorem 11). A wealth of conclusions (many of which are known by other methods) are then derived. We are able to write down a simple inversion formula for F −1 (Theorem 12). Later on in Sect. 14 we note that the inverse map F −1 depends only on very few of the coeﬃcients of F , namely only on the coeﬃcients of the

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4 face polynomials of F . We call that phenomenon The Rigidity of Morphisms (Theorem 16). A second inversion formula is given using the inverse of the logarithmic derivatives of the pair of the induced resultants of F at the origin (Theorem 13). Exact bounds on deg F −1 are given in terms of deg F . These are known even in dimensions higher than 2 and are mentioned in [2, p. 292]. The result is attributed to O. Gaber and others, however, our method of the proof is diﬀerent and yields a little sharpening of the known inequality. It gives the exact bounds on the degrees of the coordinates of F −1 in terms of the degrees of the coordinates of F (Theorem 14). Other conclusions concern the integrality of the coeﬃcients of F −1 with respect to those of F . Thus for example if the coeﬃcients of F are rational then so are the coeﬃcients of F −1 and so F (Q2 ) = Q2 (Theorem 15, Corollary 8). Section 15 gives a proof of the famous “Fibre Theorem” for maps C2 → C2 that satisfy the Jacobian condition. See Theorem 18 and compare to [14, 5]. In Sect. 16 we extract one more inversion formula and prove two more equivalent formulations to the Jacobian conjecture. This is done as follows: If F : C2 → C2 is an invertible morphism we write P (X, Y ) − α = an Y n + · · · + a0 − α = an Q(X, Y ) − β = bn Y n + · · · + b0 − β = bn

n

(Y − αi ),

i=1 n

(Y − βj )

j=1

and similarly with respect to X. We than utilize the inversion formula we found in terms of the logarithmic derivatives of the induced resultants in order to deduce the new inversion formula in terms of the zeros αi , βj (Theorem 22). This in turn leads to the new reformulations of the Jacobian conjecture (Theorem 23, Theorem 24). In Sect. 17 we address the problem of parametrizing the Jacobian variety of degree n and to calculate its dimension. This variety underlies the algorithm that solves the Jacobian conjecture for degree n or less. We give linear algebraic condition on the coeﬃcients of a polynomial P (X, Y ) ∈ C[X, Y ] in order for it to have a Jacobian mate (Theorem 25, Theorem 26). We hope that this report will motivate further related research. Hopefully, if the Jacobian conjecture is true somebody will be able to ﬁnd the algebraic relation between the Jacobian on one hand and the induced resultants on the other hand. Finally, many results given here are known in that form or another. Our bibliography is not intended to be complete and we apologize for not giving the full account and credit. To mention just a few names that we skipped: J. Stein, M. Chamberlain, G. Meisters, P. Cassou-Nogues and S. Yu. Orevkov.

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2 A Theorem of J. Hadamard We refer to the paper by J. Hadamard [9]. A copy of this paper appears also in J. Hadamard Selecta (pp. 145–159). Let f : Rn → Rn be a C map such that for any x ∈ Rn , det J(f )(x) = 0. For each x ∈ Rn we let N (x) = 1/ Df−1 (x) . Df is the diﬀerential of f and Df−1 is its inverse operator. For any R > 0 we denote μ(R) = min N (x). x=R

Theorem (J. Hadamard [9]). If f satisﬁes ∞ μ(R) dR = ∞

(1)

0

then f is a global diﬀeomorphism of Rn onto Rn . We can restate condition (1) as follows The image of any curve σ(t), 0 ≤ t < ∞, that extends to ∞ under f , i.e. f (σ(t)),

0 ≤ t < ∞,

is a non-rectiﬁable curve.

(2)

Deﬁnition 1. Let f : Rn → Rn be any map. A point x0 ∈ Rn is called an asymptotic value of f if there exists a curve σ(t), 0 ≤ t < ∞, that extends to ∞ such that lim f (σ(t)) = x0 , t→∞

where σ(t) is called an asymptotic curve of f . If we use the coordinate notations f = (f1 , . . . , fn ), x0 = (x01 , . . . , x0n ) then we call x0j an asymptotic value of the function fj , j = 1, . . . , n, and σ(t) is called an asymptotic curve of fj . Proposition 1. Let f : Rn → Rn be a C map. If f has no asymptotic values then f satisﬁes condition (2). Proof. We assume in order to get a contradiction that f does not satisfy condition (2). Let σ(t), 0 ≤ t < ∞, be a curve such that limt→∞ σ(t) 2 = ∞ but the length of f (σ(t)), 0 ≤ t < ∞, is L < ∞. We verify the Cauchy condition for f (σ(t)) ∀ > 0 ∃t() such that ∀t1 , t2 > t(), f (σ(t1 )) − f (σ(t2 )) 2 < . If not, then there is an 0 > 0 and a sequence 0 < t1 < t2 < · · · < tm → ∞ such that

f (σ(t2j )) − f (σ(t2j−1 )) 2 ≥ 0 ,

j = 1, 2, 3, . . . .

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But this implies the contradiction L≥

∞

f (σ(t2j )) − f (σ(t2j−1 )) 2 = ∞.

j=1

Hence limt→∞ f (σ(t)) = x0 exists. But this contradicts the assumption on f . Thus a corollary of Hadamard’s result is that in order to show that f is a global diﬀeomorphism it suﬃces to exclude the existence of asymptotic values for f . We will call that “Hadamard’s condition”. Remark 1. It is clear that the above deﬁnition of asymptotic values and asymptotic curves can be formulated for maps over C, i.e., F : Cn → Cn . Likewise we conclude from Hadamard’s theorem that also for the complex case it sufﬁces to exclude the existence of asymptotic values for a local diﬀeomorphism F : Cn → Cn in order to prove that it is a global diﬀeomorphism.

3 Asymptotic and Sequential Asymptotic Values of Polynomial Maps Deﬁnition 2. Let F : Cn → Cn be a map (we may formulate the deﬁnition also for maps f : Rn → Rn ). A point X0 ∈ Cn is called a sequential asymptotic n value of F if there exists a sequence {Xj }∞ j=1 in C such that limj→∞ Xj 2 = ∞ and so that lim F (Xj ) = X0 . j→∞

Remark 2. Clearly, any asymptotic value of F is a sequential asymptotic value of F . The opposite implication, however, is false. Example 1. We shall demonstrate the remark over R2 . We consider the map f : R2 → R2 , f (X, Y ) = (eX cos Y, eX sin Y ). The only asymptotic value of f is (0, 0) as is easy to check. However, any point of R2 is a sequential asymptotic value of f , for if the point (a, b) = (0, 0) this follows by the fact that (0, 0) is an asymptotic value of f while if (a, b) = (0, 0) then there is a solution Z0 to the complex equation eZ = a + ib (by Picard’s Little Theorem). Hence the following Zj = Z0 + 2πji,

j = . . . , −2, −1, 0, 1, 2, 3, . . .

are also solutions by periodicity. Since limj→∞ Zj 2 = ∞ it follows that (a, b) is a sequential asymptotic value of f .

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Notation. We will denote by A(F ) the set of all the asymptotic values of F . We will denote by AS (F ) the set of all the sequential asymptotic values of F . Remark 3. By the above we always have A(F ) ⊆ AS (F ) but in general the two sets are not the same. For polynomial maps the situation is simpler. For the proof of the next theorem we refer to the paper [19] and the preprint [18]. Theorem 1. If F : C2 → C2 is polynomial, then A(F ) = AS (F ). Proof. As in the analytic case we deﬁne a maximal domain Ω for F to be a domain in C2 , i.e., an open connected subset of C2 , with an almost smooth boundary such that F is a one to one map in Ω and Ω is maximal for inclusion with respect to these properties. Here almost smooth means smooth, except along ﬁnitely many algebraic curves. These curves are the singular locus of the mapping on the boundary. Since F is polynomial, the ﬁbre of F over any point in C2 is generically ﬁnite (unless F is constant in which case the theorem is obvious). Hence there is a tiling of C2 with ﬁnitely many maximal domains of F , {Ω1 , . . . , ΩN }. Let (a, b) ∈ AS (F ). Then there exists a sequence (Xn , Yn ) ∈ C2 such that lim (Xn , Yn ) 2 = ∞ and so that lim F (Xn , Yn ) = (a, b). We can assume that for any n, the point (Xn , Yn ) lies outside the boundaries {∂Ω1 , . . . , ∂ΩN } (by shifting a little the point if necessary and using the continuity of F ). Since the collection {Ω1 , . . . , ΩN } is ﬁnite there exists an index j, 1 ≤ j ≤ N , so that Ωj contains inﬁnitely many of the points {(Xn , Yn )}∞ n=1 . By restricting to a subsequence and re-indexing the tiles we may assume that {(Xn , Yn )}∞ n=1 ⊆ Ω1 . Hence (a, b) ∈ ∂F (Ω1 ). Since the boundary of Ω1 is almost smooth it follows that so is the boundary of F (Ω1 ). Let σ(t), 0 ≤ t ≤ 1, be a compact curve such that σ(0) = (a, b), σ(t) ∈ F (Ω1 ) for 0 < t ≤ 1 and {F (Xn , Yn )}∞ n=1 ⊆ {σ(t)|0 < t ≤ 1}. Let us pull back σ(t), 0 < t ≤ 1 via F |Ω1 . We will obtain a curve γ(t) in Ω1 which realizes (a, b) as an asymptotic value of F . Hence we obtain (a, b) ∈ A(F ).

4 The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which is the Union of Two Distinguished Algebraic Curves in C2 Deﬁnition 3. Let F : C2 → C2 , F (X, Y ) = (P (X, Y ), Q(X, Y )) be a polynomial map, i.e., P (X, Y ), Q(X, Y ) ∈ C[X, Y ]. Let α, β be two indeterminates. We will denote by R(α, β, X) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y )

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the resultant of P (X, Y ) − α, Q(X, Y ) − β considered as polynomials in Y . We have R(α, β, X) ∈ C[α, β, X]. We suppose that we have the following standard representation of this resultant as a polynomial in X R(α, β, X) = RN (α, β)X N + · · · + R0 (α, β) where for each j = 0, . . . , N we have Rj (α, β) ∈ C[α, β] and where RN (α, β) ≡ 0, i.e., degX R = N generically in α, β. Similarly we denote the resultant of P (X, Y ) − α, Q(X, Y ) − β relative to X by S(α, β, Y ) = resultant(P (X, Y ) − α, Q(X, Y ) − β, X) and assume the following standard representation S(α, β, Y ) = SM (α, β)Y M + · · · + S0 (α, β) where SM (α, β) is not the zero polynomial. The purpose of this section is to prove the following representation of the variety of the asymptotic values of a polynomial map C2 → C2 . Theorem 2. Let F : C2 → C2 , F (X, Y ) = (P (X, Y ), Q(X, Y )) be a polynomial map, then AS (F ) = {(α, β)|RN (α, β)SM (α, β) = 0}. Proof. Let (a, b) ∈ AS (F ). We will prove that RN (a, b)SM (a, b) = 0. There exists a sequence (Xn , Yn ) ∈ C2 such that lim(|Xn |2 + |Yn |2 ) = ∞ and so that lim F (Xn , Yn ) = (a, b). At least one of the coordinate sequences Xn or Yn is unbounded. Let us assume that Xn is unbounded. By restricting to a subsequence we may thus assume that lim |Xn | = ∞. We intend to show that if this is the case then RN (a, b) = 0 (similarly if the case is that lim |Yn | = ∞ then it will follow that SM (a, b) = 0). We consider P (X, Y ) − α, Q(X, Y ) − β as polynomials in Y and apply the Euclidean algorithm: (P (X, Y ) − α) = q1 (Q(X, Y ) − β) + r1 (Q(X, Y ) − β) = q2 r1 + r2 r 1 = q3 r 2 + r 3 .. . rp−2 = qp rp−1 + rp rp−1 = qp+1 rp

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where qj (α, β, X, Y ), rj (α, β, X, Y ) are polynomials in Y and rational in (α, β, X) and where degY Q > degY r1 > degY r2 > · · · . Since clearly GCD(P (X, Y ) − α, Q(X, Y ) − β) = 1 (when we consider these as polynomials in Y , because α, β are independent) we have rp ∈ C(α, β, X). Let us denote (an , bn ) = F (Xn , Yn ). Then we have the following lim an = a, lim bn = b, ∀n rp (an , bn , Xn ) = 0. Hence we have ∀n R(an , bn , Xn ) = 0. However, we are in the case where lim |Xn | = ∞ and hence we must have RN (a, b) = 0 as desired. We now have to show the inverse containment, namely, if (a, b) ∈ C2 satisﬁes RN (a, b)SM (a, b) = 0 then (a, b) ∈ AS (F ). In fact we will show more: As in the deﬁnition we let R(α, β, X) = RN (α, β)X N + · · · then we will show (i) If R(a, b, X0 ) = 0 then (a, b) ∈ F (C2 ) and {X0 } ⊆ πX F −1 (a, b). (ii) If RN (a, b) = 0 then |πX F −1 (a, b)| = N . (iii) If RN (a, b) = 0 then (a, b) ∈ AS (F ). We have R(α, β, X) = A(α, β, X, Y )(P (X, Y ) − α) + B(α, β, X, Y )(Q(X, Y ) − β) where A(α, β, X, Y ), B(α, β, X, Y ) ∈ C[α, β, X, Y ] satisfy degY A(α, β, X, Y ) < degY Q(X, Y ), degY B(α, β, X, Y ) < degY P (X, Y ). (i)

If R(a, b, X0 ) = 0 then it follows that P (X0 , Y ) − a, Q(X0 , Y ) − b have a common factor of a positive degree in Y and so (i) follows.

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(ii) If RN (a, b) = 0 then degX R(a, b, X) = N and by (i) for each one of the N zeros of R(a, b, X) there is an X-ﬁbre of F −1 (a, b) and so |πX F −1 (a, b)| ≥ N . On the other hand if X0 ∈ πX F −1 (a, b) then there is a Y0 such that F (X0 , Y0 ) = (a, b) so that R(a, b, X0 ) = 0 which proves |πX F −1 (a, b)| ≤ N. (iii) Let us pick a sequence {(an , bn )} such that lim (an , bn ) = (a, b), ∀n RN (an , bn ) = 0.

n→∞

Then for each n there are exactly N base points for the X-ﬁbres of F −1 (an , bn ). Let us denote these by πX F −1 (an , bn ) = {Xn1 , . . . , XnN }. Since RN (a, b) = 0 it follows that |πX F −1 (a, b)| < N and so by standard compactness argument there is a j, 1 ≤ j ≤ N such that limn→∞ |Xnj | = ∞. Hence (a, b) ∈ AS (F ). We can now classify the asymptotic values of a polynomial map C2 → C2 into the types X-inﬁnite and Y -inﬁnite. This should be compared to [20] where the notions of X-ﬁnite and Y -ﬁnite asymptotic values were deﬁned. Deﬁnition 4. Let F : C2 → C2 be a map (we may formulate the deﬁnition also for maps f : R2 → R2 ). A sequential asymptotic value (X0 , Y0 ) of F is 2 called X-inﬁnite (Y -inﬁnite) if there exists a sequence {(Xj , Yj )}∞ j=1 in C such that limj→∞ |Xj | = ∞ (such that limj→∞ |Yj | = ∞) and such that lim F (Xj , Yj ) = (X0 , Y0 ).

j→∞

An immediate consequence of the proof of the last theorem is the following Theorem 3. Let F : C2 → C2 , F (X, Y ) = (P (X, Y ), Q(X, Y )) be a polynomial map, then the set of all the X-inﬁnite asymptotic values of F is given by {(α, β)|RN (α, β) = 0}, and the set of all the Y -inﬁnite asymptotic values of F is given by {(α, β)|SM (α, β) = 0}. By the Bezout theorem it follows that Theorem 4. Let F : C2 → C2 , F (X, Y ) = (P (X, Y ), Q(X, Y )) be a polynomial map, then the set of asymptotic values of F which are both X-inﬁnite and Y -inﬁnite is either an inﬁnite algebraic curve or contains at most deg RN × deg SM points.

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5 The Resultant Formulation of the Jacobian Conjecture Before stating as a theorem this equivalent formulation of the Jacobian conjecture, it will be convenient to formally deﬁne few notions. Deﬁnition 5. Let n be a positive integer. A generic polynomial of degree n in the two indeterminates X, Y is a polynomial of the form aij X i Y j . P (X, Y, {aij }i+j≤n ) = i+j≤n

Thus P (X, Y, {aij }i+j≤n ) ∈ C[X, Y, {aij }i+j≤n ] and it has the following properties: (1) All the coeﬃcients of P are 1. + 2, for there are n+2 (2) The number of indeterminates of P is n+2 2 2 indeterminates of the form aij , i + j ≤ n plus X, Y . (3) P is a linear form in {aij }i+j≤n . (4) degX P = degY P = deg P = n. Deﬁnition 6. Let n be a positive integer. A generic polynomial map of degree n in the two indeterminates X, Y is an ordered pair of generic polynomials of degree n in X, Y, i.e. F = (P (X, Y, {aij }i+j≤n ), Q(X, Y, {bij }i+j≤n )), +2 indeterminates {aij }i+j≤n , {bij }i+j≤n , X, Y . The thus F involves 2 n+2 2 Jacobian of F is JF (X, Y, {aij }i+j≤n , {bij }i+j≤n ) = ∂(P, Q)/∂(X, Y ), thus JF (X, Y, {aij }i+j≤n , {bij }i+j≤n ) ∈ C[X, Y, {aij }i+j≤n , {bij }i+j≤n ] and it has the following properties: (1) (2) (3) (4)

2 The coeﬃcients of JF are integers m such that |m| ≤ n . n+2 The number of indeterminates of JF is 2 2 + 2. JF is a bilinear form in ({aij }i+j≤n , {bij }i+j≤n ). degX JF = degY JF = deg JF = 2(n − 1).

Deﬁnition 7. Let F be a generic map of degree n with the polynomial coordinates P (X, Y, {aij }i+j≤n ) and Q(X, Y, {bij }i+j≤n ). Let α, β be two new indeterminates and consider P (X, Y, {aij }i+j≤n ) − α, Q(X, Y, {bij }i+j≤n ) − β as polynomials in Y . As such, let us compute the resultant of these two polynomials. It will be denoted by resultant(P − α, Q − β, Y ).

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Thus resultant(P − α, Q − β, Y ) ∈ C[X, α, β, {aij }i+j≤n , {bij }i+j≤n ]. Similarly, let us consider P − α, Q − β as polynomials in X and as such let us compute the resultant of these two polynomials. It will be denoted by resultant(P − α, Q − β, X), thus resultant(P − α, Q − β, X) ∈ C[Y, α, β, {aij }i+j≤n , {bij }i+j≤n ]. Deﬁnition 8. Let n be a positive integer. The Jacobian variety of degree n is the variety VC (JF (X, Y, {aij }i+j≤n , {bij }i+j≤n ) − 1) where JF is considered to be a polynomial of (X, Y ). 2n = 2 equations which are bilinear Thus VC (JF − 1) is given by 2(n−1)+2 2 forms in ({aij }i+j≤n , {bij }i+j≤n ) (so in only 2 n+2 − 2 indeterminates). All 2 the equations are homogeneous except for the equation a10 b01 − a01 b10 = 1. Deﬁnition 9. Let n be a positive integer. Consider a generic polynomial map of degree n, F = (P, Q). If we consider resultant(P − α, Q − β, Y ) as a polynomial in X then each of it’s coeﬃcients is a polynomial in C[α, β, {aij }i+j≤n , {bij }i+j≤n ]. We consider these coeﬃcients as polynomials in (α, β) with coeﬃcients which are in C[{aij }i+j≤n , {bij }i+j≤n ]. The highest coeﬃcient of resultant(P − α, Q − β, Y ) is the ﬁrst coeﬃcient (highest degree of X) which is not the zero polynomial in (α, β) when it’s coeﬃcients are considered as regular functions on the Jacobian variety VC (JF −1). We shall denote the highest coeﬃcient of resultant(P − α, Q − β, Y ) by Rn (α, β, {aij }i+j≤n , {bij }i+j≤n ). Similarly, we denote the highest coeﬃcient of resultant(P − α, Q − β, X) by Sn (α, β, {aij }i+j≤n , {bij }i+j≤n ). We can now state and prove our theorem. Theorem 5. The Jacobian conjecture is equivalent to ∀n, Rn , Sn ∈ C∗ . Moreover, the Jacobian conjecture is valid for all maps of degree at most N iﬀ RN , SN ∈ C∗ . Proof. We shall prove the second equivalence which clearly implies the ﬁrst. Let N be a positive integer. The Jacobian conjecture is valid for all maps of degree at most N ⇐⇒ (by Hadamard’s condition) ∀({aij }i+j≤n , {bij }i+j≤n ) ∈ VC (JF − 1), F which is generated by ({aij }i+j≤n , {bij }i+j≤n ) has no asymptotic values ⇐⇒ (by Theorem 2) RN and SN do not vanish ⇐⇒ (by the Fundamental Theorem of Algebra) RN , SN ∈ C∗ .

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Remark 4. By symmetricity it is clear that it suﬃces to prove only RN ∈ C∗ . Example 2. F (X, Y ) = (X + (Y + X 2 )3 , Y + X 2 ). This map is tame since it factors as follows (X, Y ) → (X, Y + X 2 ) → F (X, Y ). Here ∂(P, Q)/∂(X, Y ) ≡ 1 and also resultant(P − α, Q − β, Y ) = −X + (α − β 3 ). Hence RN = −1 and it never vanishes which is consistent with Theorem 2. Example 3. F (X, Y ) = (Y 2 + XY, Y 3 + X 2 Y 2 + X 2 Y + X). This map has coordinates that belong to C[Y, XY, X 2 Y + X] = I(U −1 , V U 2 − U ), see [21, 22]. So that it must have asymptotic values. Here ∂(P, Q)/∂(X, Y ) = (3 − 4X)Y 3 − 4XY 2 − (2 + X 2 )Y − X and also resultant(P − α, Q − β, Y ) = (1 + α)X 5 + (−2 + α2 − 2α − β)X 4 + (2α + 2β + α2 )X 3 + (1 − 2αβ − 3α − 2α2 )X 2 + (−2β + 3βα)X + (β 2 − α3 ). So that RN = 1+α and hence by Theorem 3 (a, b) is an X-inﬁnite asymptotic value of F (X, Y ) iﬀ 1 + a = 0. We note that the dual map (see [20–22]) to F (X, Y ) is: G(U, V ) = ((V U 2 − U )2 + V U − 1, (V U 2 − U )3 + (V U − 1)2 + V ). Hence limU →0 F (U −1 , V U 2 − U ) = G(0, V ) = (−1, 1 + V ) so that the set {(−1, T )| T ∈ C} is contained in the variety of asymptotic values of F (X, Y ) which agrees with a = −1 above. Example 4. We shall now prove the Jacobian conjecture for maps of degree at most 2 using Theorem 5. P (X, Y ) = aX 2 + bXY + cY 2 + dX + eY = cY 2 + (bX + e)Y + (aX 2 + dX), Q(X, Y ) = AX 2 + BXY + CY 2 + DX + EY = CY 2 + (BX + E)Y + (AX 2 + DX).

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We have ∂(P, Q)/∂(X, Y ) = 2(aB − Ab)X 2 + 4(aC − Ac)XY + 2(bC − Bc)Y 2 + [2(aE − Ae) + (dB − Db)]X + [(bE − Be) + 2(dC − Dc)]Y + (dE − De). So the Jacobian variety of degree 2 is given by VC ((aB − Ab, aC − Ac, bC − Bc, 2(aE − Ae) + (dB − Db), (bE − Be) + 2(dC − Dc), dE − De − 1)). We now compute resultant(P − α, Q − β, Y ) and reduce the coeﬃcients of this polynomial (as a polynomial in X) modulo the above Jacobian variety. The result is resultant(P − α, Q − β, Y ) = (cE − eC)X + Const. Thus in our notations we have R2 = cE − eC. We need to prove that R2 ∈ C∗ . However on the Jacobian variety of degree 2 we have aB − Ab = aC − Ac = bC − Bc = 0, so if also cE − eC = 0 then this would have implied dE − De = 0. This cannot be for on the Jacobian variety we have dE −De = 1. Hence R2 = cE −eC ∈ C∗ which proves the Jacobian conjecture for all maps of degree 2 at most.

6 The Jacobian Conjecture in Dimension 2 is Decidable We are now going to generalize the last example in the sense that we shall present an algorithm that gets as an input a positive integer n and decides whether the Jacobian conjecture in dimension 2 holds true for all the polynomial maps of degree at most n. More precisely we have the following: Theorem 6. There exists an algorithm such that for any given positive integer n it decides (after ﬁnitely many steps) if the following statement is true or not. If it is not true, then the algorithm outputs a counterexample (in fact, it generates all the counterexamples). The statement is: Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and let deg P, deg Q ≤ n. If ∂(P, Q)/ ∂(X, Y ) ≡ 1 then the polynomial map F = (P, Q) is invertible. Proof. Given n we shall describe the algorithm step by step. generate the 1. Generate the Jacobian variety of degree n. To do this n+2 Step 2n equations that deﬁne the variety. These are equations in 2 − 2 inde2 2 terminates, ({aij }i+j≤n , {bij }i+j≤n ), which are determined by bilinear forms

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in those variables. All the equations are homogeneous except for the equation a10 b01 − a01 b10 = 1. Computations: aij X i Y j , P ← 1≤i+j≤n

Q←

bij X i Y j ,

1≤i+j≤n

J ← ∂(P, Q)/∂(X, Y ) =

i1 j2 (ai1 j1 bi2 j2 − ai2 j2 bi1 j1 ) X i Y j .

0≤i+j≤2(n−1) i1 +i2 −1=i j1 +j2 −1=j

So the Jacobian variety of degree n is given by the formula Vn ← VC i1 j2 (ai1 j1 bi2 j2 − ai2 j2 bi1 j1 )|1 ≤ i + j ≤ 2(n − 1) , i1 +i2 −1=i j1 +j2 −1=j

a10 b01 − a01 b10 − 1 . Step 2. Calculate resultant(P − α, Q − β, Y ). Computations:

n−j n i aij X Y j − α, P −α← j=0

Q−β ←

i=0

n−j

n j=0

bij X

i

Y j − β.

i=0

To compute the resultant we do not expand the determinant of a 2(n + 1) × 2(n + 1) matrix. Instead we use the following: Fact 1: If deg f (Y ) = n and g(Y ) = c ≡ Const. then resultant(f (Y ), g(Y ), Y ) = resultant(f (Y ), c, Y ) = Cn . Fact 2: If deg f (Y ) = n and deg g(Y ) = m then resultant(f (Y ), g(Y ), Y ) = (−1)nm resultant(g(Y ), f (Y ), Y ). Fact 3: If f = An Y n + · · · + A0 and g = Bm Y m + · · · + B0 and m ≤ n we let h = f − (An /Bm )Y n−m g,

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and then we have n−deg h resultant(h(Y ), g(Y ), Y ). resultant(f (Y ), g(Y ), Y ) = Bm

We note that deg h ≤ n − 1. Fact 4: Follows from Fact 3: If we use the division algorithm to write f = qg + r where deg r < deg g then n−deg r resultant(r(Y ), g(Y ), Y ). resultant(f (Y ), g(Y ), Y ) = Bm

We can now iterate using Facts 4 and 2 in order to reduce degrees. For example if using the division algorithm we write f = q1 g + r1 , g = q2 r1 + r2 then deg f −deg r1 resultant(r1 , g, Y ) resultant(f, g, Y ) = Bm deg g deg r1 deg f −deg r1 = (−1) Bm resultant(g, r1 , Y ) deg f −deg r1 deg g−deg r2 = (−1)deg g deg r1 Bm Cl × resultant(r2 , r1 , Y ) deg f −deg r1 deg g−deg r2 = (−1)deg g deg r1 +deg r1 deg r2 Bm Cl × resultant(r1 , r2 , Y ),

where Bm , Cl are the leading coeﬃcients of g, r1 . To put that in a form of a pseudocode: Input: f, g Output: resultant(f, g, Y ) h←f s←g res ← 1 WHILE deg s > 0 DO r ← remainder (h, s) Res ← (−1)deg r deg s lead (s)deg h−deg r Res h←s s←r IF h = 0 OR s = 0 THEN Res ← 0 ELSE IF deg h > 0 THEN Res ← sdeg h Res END Step 3. Reduce the coeﬃcients of resultant(P − α, Q − β, Y ) modulo the Jacobian variety of degree n, Vn . Start with the highest coeﬃcient and proceed till you get to the ﬁrst coeﬃcient Rn which is diﬀerent from 0. Computations: By Step 1 we have

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Vn ← VC

i1 j2 (ai1 j1 bi2 j2 − ai2 j2 bi1 j1 )|1 ≤ i + j ≤ 2(n − 1) ,

i1 +i2 −1=i j1 +j2 −1=j

a10 b01 − a01 b10 − 1 . Find a Gr¨obner basis Gn for the ideal generated by the deﬁning equations of Vn . Consider the coeﬃcient CN of X N in resultant(P − α, Q − β, Y ). CN is a polynomial in (α, β, {aij }1≤i+j≤n , {bij }1≤i+j≤n ). Express CN =

Aij ({akl }, {bkl })αi β j ,

where Aij ∈ C[{aij }i+j≤n , {bij }i+j≤n ]. For each (i, j) divide the polynomial Aij by the Gr¨obner basis Gn = {g1 , . . . , gt } of degree n to get Aij = q1 g1 + · · · + qt gt + rij . We have Aij ≡ 0 (mod Vn )

⇐⇒

rij ≡ 0.

This algorithm, obviously, terminates after ﬁnitely many steps. If Rn ∈ C∗ then the Jacobian conjecture is established up to (including) degree n. Otherwise it is clear how to generate all the counterexamples for degree n. Remark 5. For Step 3, references to the Buchberger algorithm for computing Gr¨ obner basis are [3, 4, 8]. The complexity of the Buchberger type algorithms is huge. There are some results on uniform upper bounds on the degrees of the Gr¨ obner basis elements in terms of the original generators [4, 13, 16]. The bounds on the degrees are large. That is a necessity in some sense, for in [13] there are examples of ideals generated by polynomials of degree n at most, whose Gr¨ obner basis involve polynomials of degree proportional to n 22 . Remark 6. The algorithm above was implemented by the author (while visiting the University of Michigan) on a Sun platform. He used the standard Gr¨ obner package in Maple. The input to the program was a positive integer d and the output was a true/false ﬂag. If true then the Jacobian conjecture was found to be valid for all the complex polynomial mappings in dimension two and of degree d or less. A false ﬂag meant the other alternative and a counterexample of degree d or less would have been given. The package was able to automatically prove the conjecture for mappings of degree 15 or less. The running time was about 40 h. It was instrumental to observe how fast the running time of the program went up with the degree of the polynomials. For low degrees (up to degree 5) it ﬁnished within less than 5 min. Increasing

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the degree further had a very signiﬁcant eﬀect on the elapsed running time. It quickly climbed to about one hour in degree 7 to about three hours up to degree 10. It than went to an overnight running time in degree 13 and degrees 14, 15 quickly reached the few days running time. The author did not check the memory size that was involved and it is quite possible that the elapsed running time had a signiﬁcant portion due to memory paging. It hints that very quickly one expects the combination of the max degree of the reduced Gr¨ obner basis and its size (in terms of how many polynomials it contains) to get very large. Remark 7. After the Linz conference Massimiliano (Max) Sala went into programming the algorithm more professionally. It is now an ongoing project of Max and his students. The project started while Max was in Ireland and it continues now in 2009 while he is in Trento, Italy. In the very ﬁrst experiments Max used the degrevlex. For n = 2 the Jacobian ideal was a radical. It had 13 elements, 9 of degree 2 and 4 of degree 3. The 9 degree 2 polynomials formed a minimal basis and so the 4 degree 3 polynomials were inessential. The largest coeﬃcient was 4 which was a good sign (usually the coeﬃcients of the members of the basis might get very large). The 9 polynomials seemed to be sparse. Four of them had very nice structure: b211 − 4b20 b02 , a211 − 4a20 a02 , a11 b11 − 4a20 b02 and a02 b20 − a20 b02 . However, already in degree n = 3 the Gr¨ obner basis had about 3000 polynomials and much less structure was visible.

7 A Straight Forward Inductive Approach Fails The results of the previous section motivate an inductive approach to deal with the Jacobian conjecture. The purpose of this section is to describe the simplest possible such a trial and to show that it does not work. This may indicate a negative evidence for the validity of the conjecture in dimension two. Let n > 1 be an integer. We are going to make the following induction hypothesis: Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] satisfy (1) degY P = degY Q = deg P = deg Q = n − 1, (2) ∂(P, Q)/∂(X, Y ) ∈ C∗ . Then

Rn−1 ∈ C∗ .

We now proceed to the proof of the following: Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] satisfy (1) degY P = degY Q = deg P = deg Q = n, (2) ∂(P, Q)/∂(X, Y ) ∈ C∗ .

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Then P (X, Y ) = cY n + · · · ,

Q(X, Y ) = CY n + · · ·

for some c, C ∈ C∗ . Let P1 (X, Y ) = CP (X, Y ) − cQ(X, Y ),

Q1 (X, Y ) = cQ(X, Y ).

Then P1 (X, Y ) = C[cY n + (bX + e)Y n−1 + · · · ] − c[CY n + (BX, E)Y n−1 + · · · ] = [(Cb − cB)X + (Ce − cE)]Y n−1 + · · · . Now

bY n−1 + · · · + ncY n−1 + · · · ∂(P, Q)/∂(X, Y ) = BY n−1 + · · · + nCY n−1 + · · · = n(bC − Bc)Y 2(n−1) + · · · + ∈ C∗ .

So that Cb − cB = 0 and we obtain P1 (X, Y ) = (Ce − cE)Y n−1 + · · · . We note that ∂(P1 , Q1 )/∂(X, Y ) = ∂(CP − cQ, cQ)/∂(X, Y ) = ∂(CP, cQ)/∂(X, Y ) = cC∂(P, Q)/∂(X, Y ) ∈ C∗ . Also resultant(P1 − α, Q1 − β, Y ) = resultant(CP − cQ − α, cQ − β, Y ) = {(−1)n /cC}resultant(CP − (α + β), cQ − β, Y ). This shows that RnP1 ,Q1 (α, β) = {(−1)n /cC}RnCP,cQ (α + β, β). Here RnF denotes the highest coeﬃcient of the Y -resultant that corresponds to the map F . We have degY P1 ≤ n − 1 so we could have used the induction hypothesis if degY Q1 ≤ n−1. However degY Q1 = degY Q = n unfortunately. Looking at P1 (X, Y ) = (Ce − cE)Y n−1 + · · · , Q1 (X, Y ) = cCY n + c(BX + E)Y n−1 + · · · , and assuming for a moment that Ce − cE ∈ C∗ it could have made sense to divide Q1 by P1 and consider:

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P2 (X, Y ) = cCP1 (X, Y ), Q2 (X, Y ) = (Ce − cE)Q1 (X, Y ) − cCP1 (X, Y )Y, for then we would get degY Q2 ≤ n − 1 and the resultants of (P1 , Q1 ) and of (P2 , Q2 ) would have diﬀer by a constant non-zero factor. However, this transformation destroys the Jacobian condition for ∂(P2 , Q2 )/∂(X, Y ) = cC(Ce − cE)∂(P1 , Q1 )/∂(X, Y ) − (cC)2 P1 P1X . Going back to P1 (X, Y ) = (Ce − cE)Y n−1 + · · · , Q1 (X, Y ) = cCY n + c(BX + E)Y n−1 + · · · , we might look for a more general transformation on Q1 (X, Y ) than just the division transform by P1 . Namely, we might want to ﬁnd a polynomial S(X, Y ) = −cCY n + · · · such that ∂(P1 , S)/∂(X, Y ) ≡ 0 for then if we consider P2 (X, Y ) = P1 (X, Y ), Q2 (X, Y ) = Q1 (X, Y ) − S(X, Y ), then also degY Q2 ≤ n − 1 but this time the Jacobian condition is preserved for ∂(P2 , Q2 )/∂(X, Y ) = ∂(P1 , Q1 + S)/∂(X, Y ) = ∂(P1 , Q1 )/∂(X, Y ) + ∂(P1 , S)/∂(X, Y ) = ∂(P1 , Q1 )/∂(X, Y ) ∈ C∗ . P2 ,Q2 P1 ,Q1 There is also a hope to ﬁnd a relation between Rn−1 and Rn−1 . However, there is no hope in ﬁnding such a S(X, Y ) except in the case n = 2 in which the Jacobian conjecture is true. The reason lies in the fact that if P (X, Y ) ∈ C[X, Y ] and if we deﬁne the following derivation on C[X, Y ]

DP : C[X, Y ] → C[X, Y ], DP (g) = ∂(P, g)/∂(X, Y ), then, as we shall see in the next four sections we have ker(DP , C[X, Y ]) = C[P ]. Thus if P (X, Y ) = Y +· · · and S(X, Y ) = Y n +· · · satisfy DP (S) = 0 then S(X, Y ) = G(P (X, Y )) for some G(T ) ∈ C[T ] so Y n + · · · = G(Y n−1 + · · · ) which implies that n = (n − 1) deg G n−1

so that n = deg G = 2 is the only solution. It is indeed a solution for we take G(T ) = T 2 and S = P 2 . We will discuss this property of the Jacobian derivation DP as well as other properties on the next four sections.

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8 Elementary Properties of Resultants of Jacobian Pairs Since we have a resultant formulation for the Jacobian conjecture it makes sense to establish some properties of resultants of Jacobian pairs. This section is dedicated to the more elementary such properties. Proposition 2. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] be a Jacobian pair. Let α and β be indeterminates.Then the highest coeﬃcient of resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) cannot be the coeﬃcient of X 0 . Proof. Assume that the proposition is false. Then resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = R0 (α, β) ∈ C[α, β]. We consider 2 cases: Case 1: R0 (α, β) ∈ C∗ . Then for any pair (α, β) the equations P (X, Y ) − α = 0, Q(X, Y ) − β = 0, have no solution which is a contradiction. Case 2: Let (α0 , β0 ) ∈ C2 satisfy R0 (α0 , β0 ) = 0. Then for any X there exists a Y (X) such that P (X, Y (X)) = α0 , Q(X, Y (X)) = β0 . Diﬀerentiations of these equations with respect to X give PX (X, Y (X)) + Y (X)PY (X, Y (X)) = 0, QX (X, Y (X)) + Y (X)QY (X, Y (X)) = 0. We consider these as two homogeneous equations in (1, Y (X)). Then we get PX (X, Y (X))PY (X, Y (X)) =0 QX (X, Y (X))QY (X, Y (X)) which contradicts the Jacobian condition.

Remark 8. We shall call the argument that was used in the proof of case 2 above, “the level curve principle”. It is merely the obvious fact that a Jacobian pair cannot share a common level curve. Remark 9. In case 2 above we could have argued diﬀerently: If R0 (α, β) ≡ 0 then P (X, Y ) − α, Q(X, Y ) − β had to have a common factor of a positive degree in Y . That is clearly impossible for α, β are independent (also because P (X, Y ) − α, Q(X, Y ) − β is a Jacobian pair so these polynomials must be coprime).

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If deg R0 (α, β) > 0 then the equations P (X, Y ) − α = 0, Q(X, Y ) − β = 0, had a solution only along the algebraic curve R0 (α, β) = 0. So the image of the map F (X, Y ) = (P (X, Y ), Q(X, Y )) collapses to the algebraic curve R0 (α, β) = 0. That cannot be for F is ´etale and hence in particular an open map. Proposition 3. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] be a Jacobian pair. Let α and β be indeterminates. Then there cannot exist a pair (α0 , β0 ) such that ∀X resultant(P (X, Y ) − α0 , Q(X, Y ) − β0 , Y ) ≡ 0. Proof. Assume that there exists such a pair (α0 , β0 ) ∈ C2 . Then we use, as before, the level curve principle to arrive at a contradiction, namely: for any X there exists a Y (X) such that P (X, Y (X)) = α0 , Q(X, Y (X)) = β0 , and we arrive at a contradiction. Remark 10. This proposition generalizes case 2 in the proof of Proposition 2. Corollary 1. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] be a Jacobian pair. Let α and β be indeterminates. Let resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = RN (α, β)X N + · · · + R0 (α, β), where RN (α, β) is the highest non-vanishing (identically) coeﬃcient of resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ). Then Rj (α, β), 0 ≤ j ≤ N , cannot share a zero. In particular the highest common divisor of (RN (α, β), . . . , R0 (α, β)) is 1. Corollary 2. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] be a Jacobian pair. Let α and β be indeterminates. Let resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = R1 (α, β)X + R0 (α, β), where R1 (α, β) is the highest non-vanishing (identically) coeﬃcient of resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ). Then the image of the map F (X, Y ) = (P (X, Y ), Q(X, Y )) is C2 − {(α, β)|R1 (α, β) = 0}.

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Proof. If R1 (α0 , β0 ) = 0 then by the previous corollary R0 (α0 , β0 ) = 0 so that the resultant is not 0 and the equations P (X, Y ) = α0 , Q(X, Y ) = β0 , have no solution. If, on the other hand, R1 (α0 , β0 ) = 0 then X0 = −R0 (α0 , β0 )/R1 (α0 , β0 ) is the zero of the resultant and so there exists a Y0 such that P (X0 , Y0 ) = α0 , Q(X0 , Y0 ) = β0 . So (α0 , β0 ) ∈ F (C2 ).

Corollary 3. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] be a Jacobian pair. Let α and β be indeterminates. If resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = R1 X + R0 (α, β), where R1 ∈ C∗ then F (X, Y ) = (P (X, Y ), Q(X, Y )) is onto C2 . In fact we can do better than in the last two corollaries. Let us prove the following covering theorem. Theorem 7. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and denote F (X, Y ) = (P (X, Y ), Q(X, Y )). Then there are ﬁnitely many polynomials RN (α, β), . . . , R1 (α, β) ∈ C[α, β] such that C2 − F (C2 ) = {(a, b) ∈ C2 |RN (a, b) = · · · = R1 (a, b) = 0} = VC (RN , . . . , R1 ). Moreover, C2 − F (C2 ) is inﬁnite iﬀ ∂(P, Q)/∂(X, Y ) ≡ 0, provided that the highest coeﬃcients of X, Y are non-zero constants. Remark 11. This theorem is the analog of Picard’s Little Theorem for analytic functions. Proof. Let α, β be two new indeterminates and let us form R(X) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) ∈ C[X, α, β]. Suppose that we have the following representation of R(X) as a polynomial in X R(X) = RN (α, β)X N + · · · + R1 (α, β) + R0 (α, β), where RN (α, β), . . . , R0 (α, β) ∈ C[α, β]. Then we have (a, b) ∈ C2 −F (C2 ) ⇐⇒ there is no solution in (X, Y ) to P (X, Y ) − a = Q(X, Y ) − b = 0 ⇐⇒ for any X0 , R(X0 ) = 0 where (α, β) ← (a, b) ⇐⇒ R(X) ∈ C∗ where (α, β) ← (a, b)

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⇐⇒ R(X) = RN (a, b)X N + · · · + R0 (a, b) = R0 (a, b) = 0 ⇐⇒ RN (a, b) = · · · , R1 (a, b) = 0 Moreover, by the Bezout Theorem C2 − F (C2 ) is inﬁnite ⇐⇒ L(α, β) = (RN (α, β), . . . , R1 (α, β)) ∈ C[α, β] − C∗ ⇐⇒ L(P (X0 , Y0 ), Q(X0 , Y0 )) = 0 ∀(X0 , Y0 ) ⇐⇒ L(P (X, Y ), Q(X, Y )) = c ∈ C∗ ⇐⇒ ∂(P, Q)/∂(X, Y ) ≡ 0. Corollary 4. If degX resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = 1, then either ∂(P, Q)/∂(X, Y ) ≡ 0 or F (X, Y ) = (P (X, Y ), Q(X, Y )) is onto (for R1 (α, β) = 0 is either empty or inﬁnite over C). Remark 12. We already know that if the Jacobian conjecture is true then the Jacobian condition should imply that Rn (α, β) ∈ C∗ where Rn (α, β) is the highest coeﬃcient of resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ). It is natural to ask if the following more concrete statement holds true: If RN (α0 , β0 ) = 0 then there exists (X0 , Y0 ) such that P (X0 , Y0 ) = α0 , Q(X0 , Y0 ) = β0 , for which ∂(P, Q)/∂(X, Y )(X0 , Y0 ) = 0? The answer is negative. Example 5. We have considered this example before: P (X, Y ) = Y 2 + XY,

Q(X, Y ) = Y 3 + X 2 Y 2 + X 2 Y + X.

Then we computed resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = (1 + α)X 5 + · · · . Thus R5 (−1, β0 ) = 0 for any β0 ∈ C and we need to check if ∂(P, Q)/∂(X, Y )(X0 , Y0 ) = 0 for a point (X0 , Y0 ) for which P (X0 , Y0 ) = Y02 + X0 Y0 = −1, Q(X0 , Y0 ) = Y03 + X02 Y02 + X02 Y0 + X0 = β0 , by the ﬁrst equation we have Y02 = −1 − X0 Y0 . So using the second equation we get Y03 = β0 1/3

so Y0 = β0 are the three possible solutions and X0 = −(1 + Y02 )/Y0 or −1/3 1/3 − β0 are the three corresponding X0 ’s. X0 = −β0

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We now compute −1/3

1/3

1/3

∂(P, Q)/∂(X, Y )(−β0 − β0 , β0 ) 3 2 = (3 − 4X)Y − 4XY − (2 + X 2 )Y − X|(X0 ,Y0 ) 4/3

= 4β0

2/3

+ 6β0 + 4β0

1/3

+ β0 .

Since β0 = 0 we see that in general this is not zero. It vanishes exactly for 1/3 T0 = β0 which satisﬁes: 4T03 + 6T02 + 4T0 + 1 = 0. Thus, maybe, only the following is true: Conjecture: ∂(P, Q)/∂(X, Y ) has zero on F −1 (Rn (α, β) = 0). This is an equivalent formulation for the Jacobian conjecture because of the Hadamard condition on a local diﬀeomorphism to be a global one, and because for a polynomial automorphism the leading coeﬃcient Rn (α, β) belongs to C× . However, this formulation is sharper than the standard formulation in that it gives a location for a zero of the determinant of the Jacobian in the case of a polynomial mapping which is not an automorphism.

9 Grading an Algebra with a Derivation – Introduction This section serves as an introductory section for the next three sections. If k is a ﬁeld of characteristic 0 and A is a k-algebra with 1 and D a derivation of A, then there is a natural gradation of A with respect to D: the elements of class n in the gradation are those that are annihilated by n iterations of D. If we assume, further, the existence of an element q ∈ A for which D(q) = 1 then in “reasonable” algebras the multiplication of an element in class n by q is an element of class n + 1 and conversely (Theorem 8). As a consequence we obtain a structure theorem for the classes of this D-gradation in terms of ker(D, A) and of q (Theorem 9). This leads immediately to the well known fact that Nil (D) = ker(D, A)[q] which was noted by several people [23, 26] and [17] (Corollary 1). It also gives a necessary and suﬃcient condition for D to be a locally nilpotent derivation of A (Corollary 6). One application of the above is for the polynomial ring over k in n variables, A = k[x1 , . . . , xn ]. Given n elements f1 , . . . , fn ∈ A that satisfy the Jacobian condition, namely, that ∂(f1 , . . . , fn )/∂(x1 , . . . , xn ) = 1 one can take D = ∂(f1 , . . . , fn−1 , .)/∂(x1 , . . . , xn ), q = fn and apply the above. This is closely related to the results on the derivations d1 , . . . , dn that were introduced in [17], however, the methods of proofs are diﬀerent.

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10 Grading an Algebra According to a Derivation Let k be a ﬁeld of characteristic 0, A a k-algebra with 1 and D a derivation of A. Deﬁnition 10. For n ∈ Z+ {0} we denote An (D) = {f ∈ A | Dn (f ) = 0}. Remark 13. By the deﬁnition we have A0 (D) = {0}, A1 (D) = ker(D, A). Also An (D) ⊆ An+1 (D) for any n ∈ Z+ {0}. ∞ Deﬁnition 11. A∞ (D) = A − n=0 An (D). Let q ∈ A satisfy D(q) = 1. In this section we shall see that, roughly, the action of multiplication by q on the classes An (D) is to advance the class index by 1. It will be convenient to state two lemmas ﬁrst. Lemma 1. For any f ∈ A and for any n ∈ Z+ {0} we have (n + 1)Dn (f ) = Dn+1 (f q) − Dn+1 (f )q. Proof. By D(f q) = D(f )q + f D(q) = D(f )q + f we get f = D(f q) − D(f )q which takes care of the case n = 0. We proceed by induction on n. We assume that nDn−1 (f ) = Dn (f q) − Dn (f )q, n ≥ 1. Then D(nDn−1 (f )) = D(Dn (f q) − Dn (f )q), nDn (f ) = Dn+1 (f q) − Dn+1 (f )q − Dn (f ). So (n + 1)Dn (f ) = Dn+1 (f q) − Dn+1 (f )q.

Lemma 2. If f ∈ A satisﬁes Dn+1 (f q) = 0 then for every k ∈ Z+ , Qk is a right divisor of Dn (f ). Proof. We will see by induction on k that (n + k)Dn+k−1 (f ) = −Dn+k (f )q. By Dn+1 (f q) = 0 and by Lemma 1 we have the case k = 1. We now assume that (n + k)Dn+k−1 (f ) = −Dn+k (f )q. Then D((n + k)Dn+k−1 (f )) = D(−Dn+k (f )q), (n + k)Dn+k (f ) = −Dn+k+1 (f )q − Dn+k (f ). So (n + k + 1)Dn+k (f ) = −Dn+k+1 (f )q. This identity shows that (n + 1)Dn (f ) = (−1)k Dn+k (f )Qk /(n + 2)(n + 3) · · · (n + k).

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We are now ready to prove the relation between multiplication by q and the D-gradation of A. Theorem 8. (i) If f ∈ A satisﬁes Dn (f ) = 0 and Dn−1 (f ) = 0 then Dn+1 (f q) = 0 and Dn (f q) = 0. (ii) If for any g ∈ A − {0} there exists an n(g) such that Qn(g) is not a right divisor of g, then if f ∈ A satisﬁes Dn+1 (f q) = 0 and Dn (f q) = 0 then Dn (f ) = 0 and Dn−1 (f ) = 0. Proof. (i) By Dn (f ) = 0 we get Dn+1 (f ) = 0 and so Lemma 1 implies that Dn+1 (f q) = 0. Also by Lemma 1 nDn−1 (f ) = Dn (f q) and so Dn (f q) = 0. (ii) By Lemma 2 Dn (f ) = 0. By Lemma 1 nDn−1 (f ) = Dn (f q) and so Dn−1 (f ) = 0.

11 The Structure of the D-classes We now give a structure theorem for the classes An (D) in terms of ker(D, A) and of q (it has been noted before, see e.g. [23, 26] and [17]). Our proof is completely elementary. Deﬁnition 12. Let B be a k-algebra with 1. For n ∈ Z+ we will denote B (n) [q] = {F (q) | F (x) ∈ B[x], deg F < n}. Theorem 9. An (D) = (A1 (D))(n) [q] = (ker(D, A))(n) [q]. Proof. The assertion holds trivially for n = 1. We proceed by induction on n. We assume that Am (D) = (A1 (D))(m) [q] for 1 ≤ m ≤ n. By Theorem 8(i) we have the inclusion (A1 (D))(n+1) [q] = A1 (D) + An (D)q ⊆ An+1 (D). For the inverse inclusion we take an f ∈ An+1 (D). We compute n (−1)m+1 Dm (f )Qm /m! D f− m=1

= D(f ) − = D(f ) − n

n m=1 n

(−1)m+1 D(Dm (f )Qm )/m! (−1)m+1 (mDm (f )Qm−1 + Dm+1 (f )Qm )/m! = 0.

m=1

Hence f − m=1 (−1)m+1 Dm (f )Qm /m! ∈ A1 (D). Also by the induction hy pothesis Dm (f ) ∈ (A1 (D))(n−m+1) [q] and so f ∈ (A1 (D))(n+1) [q].

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Corollary 5.

ker(D, A)[q] = A1 (D)[q] = f ∈ A | ∃n ∈ Z+ {0}, Dn (f ) = 0 .

Corollary 6. D is a locally nilpotent derivation of A iﬀ A1 (D)[q] = ker(D, A)[q] = A. Corollary 7. A∞ (D) = A − A1 (D)[q] = A − ker(D, a)[q].

12 Application to Automorphisms of Polynomial Rings We apply the results of the previous section to derive some properties of automorphisms of polynomial rings (in characteristic 0). These are strongly related to Theorem 3.3 in [17]. Let k be, as before, a ﬁeld of characteristic 0. We will take A = k[x1 , . . . , xn ] the polynomial algebra in n variables over k. Any n-tuple (f1 , . . . , fn ) ∈ An determines a k-endomorphism ϕ = ϕ(f1 , . . . , fn ) of A by ϕ(xj ) = fj , 1 ≤ j ≤ n. Also the operator D(f1 ,...,fn−1 ) : A −→ A, D(f1 ,...,fn−1 ) (g) = ∂(f1 , . . . , fn−1 , g)/∂(x1 , . . . , xn ), is a k-derivation of A which we call the (f1 , . . . , fn−1 )-Jacobian derivation. Clearly we have A1 (D(f1 ,...,fn−1 ) ) = ker(D(f1 ,...,fn−1 ) , A) ⊇ k[f1 , . . . , fn−1 ], for n = 2 the inclusion is known to be an equality. Theorem 10. Let k be a ﬁeld of characteristic 0 and let f1 , . . . , fn ∈ k[x1 , . . . , xn ] satisfy D(f1 ,...,fn−1 ) (fn ) = 1. (i) If ϕ(f1 , . . . , fn ) is invertible then ker(D(f1 ,...,fn−1 ) , A) = k[f1 , . . . , fn−1 ] and D(f1 ,...,fn−1 ) is a locally nilpotent derivation of k[x1 , . . . , xn ]. (ii) If n = 2 then ϕ(f1 , f2 ) is invertible iﬀ D(f1 ) is a locally nilpotent derivation of k[x1 , x2 ]. Proof. (i) ϕ(f1 , . . . , fn ) is invertible iﬀ k[f1 , . . . , fn ] = k[x1 , . . . , xn ]. On the other hand, as noted above A1 (D(f1 ,...,fn−1 ) ) ⊇ k[f1 , . . . , fn−1 ]. ⊇ k[f1 , . . . , fn ]. Thus we have Hence A1 (D(f1 ,...,fn−1 ) )[fn ] A1 (D(f1 ,...,fn−1 ) ) = k[f1 , . . . , fn−1 ] and so by Corollary 6 D(f1 ,...,fn−1 ) is locally nilpotent. (ii) For n = 2 we have A1 (D(f1 ) ) = k[f1 ] and so we have the desired equivalence.

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13 Invertible Morphisms, Their Resultants and Inversion Formulas Some of the results of this section are well known results. The purpose of this section is to bring up all these results in a uniﬁed manner with the aid of the notion of the resultant of a map. Also we will discover some new results, as well. Theorem 11. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism. Then there exist two positive integers N and M such that resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = RN (X + r0 (α, β))N , resultant(P (X, Y ) − α, Q(X, Y ) − β, X) = SM (X + s0 (α, β))M , where RN , SM ∈ C∗ and where r0 (α, β), s0 (α, β) ∈ C[α, β]. Proof. Let resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = RN X N + RN −1 (α, β)X N −1 + · · · + R0 (α, β). Since F (X, Y ) is a morphism it follows that P (X, Y ), Q(X, Y ) is a Jacobian pair so by the results on the previous section, N ≥ 1. Also since F (X, Y ) cannot have asymptotic values it follows by the resultant formulation of the Jacobian conjecture that RN ∈ C∗ . Clearly, RN −1 (α, β), . . . , R0 (α, β) ∈ C[α, β]. Given any (α0 , β0 ) and any X0 which is a zero of RN X N + · · · + R0 (α0 , β0 ),

(3)

there exists a Y0 such that F (X0 , Y0 ) = (α0 , β0 ). Since F (X, Y ) is injective it follows that all the zeros of (3) must coincide. This proves that RN X N + · · · + R0 (α0 , β0 ) = RN (X N + RN −1 (α, β)/RN X N −1 + · · · + R0 (α, β)/RN ) = RN (X + (R0 (α, β)/RN )1/N )N . In particular we obtain the following N + 1 relations N Rj (α, β)/RN = (R0 (α, β)/RN )(N −j)/N , j

0 ≤ j ≤ N.

(4)

(5)

In particular for j = N − 1 we get (R0 (α, β)/RN )1/N = RN −1 (α, β)/(N RN ) ∈ C[α, β]. A similar argument proves the result for resultant(P (X, Y ) − α, Q(X, Y ) − β, X).

(6)

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More Analysis: Let us assume that we have the following standard representations P (X, Y ) = an (X)Y n + · · · + a0 (X), Q(X, Y ) = bm (X)Y

m

degY P (X, Y ) = n,

+ · · · + b0 (X),

degY Q(X, Y ) = m.

(7)

Then by the Sylvester’s formula we get resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) an (X) ... 0 bm (X) .. .. . ... an (X) . .. = a0 (X) − α . . . . b0 (X) − β .. .. .. . ... . . 0 . . . a0 (X) − α 0 So in particular we have an (0) .. . R0 (α, β) = a0 (0) − α .. . 0

... ... ... ... ...

...

0

bm (0)

...

...

an (0)

.. .

...

...

.. .

b0 (0) − β

...

...

.. .

.. .

...

...

a0 (0) − α

0

...

bm (X) .. . (8) . .. . b0 (X) − β 0

bm (0) .. . . .. . b0 (0) − β 0

(9)

Hence deg R0 (α, β) ≤ max(n, m) or equivalently deg R0 (α, β) ≤ max(degY P (X, Y ), degY Q(X, Y )) = degY F (X, Y ).

(10)

By (4) we obtain the following conclusion: If P (X, Y ) = α, Q(X, Y ) = β then X(α, β) = −(R0 (α, β)/RN )1/N . By (6) X(α, β) ∈ C[α, β] and by (10) deg X(α, β) ≤ degY F (X, Y ). Similarly if resultant(P (X, Y ) − α, Q(X, Y ) − β, X) = SM Y M + SM −1 (α, β)Y M −1 + · · · + S0 (α, β), then we have: SM Y M + · · · + S0 (α, β) = SM (Y + (S0 (α, β)/SM )1/M )M , M (S0 (α, β)/SM )(M −j)/M , 0 ≤ j ≤ M, Sj (α, β)/SM = j

(11)

(S0 (α, β)/SM )1/M = SM −1 (α, β)/(M SM ) ∈ C[α, β],

(13)

(12)

deg S0 (α, β) ≤ max(degX P (X, Y ), degX Q(X, Y )) = degX F (X, Y ). (14)

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So by (11) we obtain the following conclusion: If P (X, Y ) = α Q(X, Y ) = β then Y (α, β) = −(S0 (α, β)/SM )1/M . By (13) Y (α, β) ∈ C[α, β] and by (14) deg Y (α, β) ≤ degX F (X, Y ). Remark 14. It is possible to show that N = M = 1. We now can conclude more results that emerge from the proof of the last theorem. Theorem 12. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism. Let us assume that we have the following standard representations: P (X, Y ) = an (X)Y n + · · · + a0 (X) = AN (Y )X N + · · · + A0 (Y ), where degY P (X, Y ) = n and degX P (X, Y ) = N . Q(X, Y ) = bm (X)Y m + · · · + b0 (X) = BM (Y )X M + · · · + B0 (Y ), where degY Q(X, Y ) = m and degX Q(X, Y ) = M . Let F −1 (α, β) = (X(α, β), Y (α, β)). Then there are R1 , S1 ∈ C∗ so that we have the following formulas for the inverse map an (0) ... 0 bm (0) ... 0 .. .. . . . . a (0) . . . . b (0) n m .. .. −R1 X(α, β) = a0 (0) − α . . . , . b0 (0) − β . . . . .. .. .. .. . ... . . ... . 0 . . . a0 (0) − α 0 . . . b0 (0) − β and AN (0) .. . −S1 Y (α, β) = A0 (0) − α .. . 0

...

0

BM (0)

...

...

AN (0)

.. .

...

...

.. .

B0 (0) − β

...

...

.. .

.. .

...

...

A0 (0) − α

0

...

Proof. This follows by the discussion after (10) and by (9).

BM (0) .. . . .. . B0 (0) − β 0

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Example 6. Degree 1. P (X, Y ) = aX + bY, b B R(α, β) = = Bα − bβ, −α −β

Q(X, Y ) = AX + BY, A a S(α, β) = = aβ − Aα. −β −α

On the other hand calculation of the inverse gives: aX + bY = α, AX + BY = β =⇒ X = (aB − bA)−1 R(α, β), Y = (aB − bA)−1 S(α, β) Example 7. Degree 2. P (X, Y ) = 4Y 2 + (1 + 4X)Y + X 2 + X = X 2 + (1 + 4Y )X + 4Y 2 + Y, Q(X, Y ) = −4Y 2 + (1 − 4X)Y − X 2 = −X 2 − 4XY − 4Y 2 + Y, 4 0 −4 0 1 4 1 −4 R(α, β) = = 16α2 + 32αβ + 16β 2 − 8α + 8β. −α 1 −β 1 0 −α 0 −β So R(α, β) = 16(α + β)2 − 8(α − β). Similarly we have S(α, β) = (α + β)2 + β. On the other hand calculation of the inverse gives: α = X + Y + (X + 2Y )2 , =⇒

β = Y − (X + 2Y )2 , X = −8R(α, β), Y = S(α, β).

Theorem 13. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism. Let us denote F −1 (α, β) = (X(α, β), Y (α, β)). Then we have the following formulas for the inverse map X(α, β) = −R(X, α, β)/(d/dX{R(X, α, β)})(0), Y (α, β) = −S(Y, α, β)/(d/dY {S(Y, α, β)})(0), where R(X, α, β) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ), S(Y, α, β) = resultant(P (X, Y ) − α, Q(X, Y ) − β, X). Proof. This follows by the previous theorem and by (9).

Theorem 14. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism. Let us denote F −1 (α, β) = (X(α, β), Y (α, β)). Then

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deg X(α, β) ≤ max(degY P (X, Y ), degY Q(X, Y )) = degY F (X, Y ), deg Y (α, β) ≤ max(degX P (X, Y ), degX Q(X, Y )) = degX F (X, Y ), deg P (X, Y ) ≤ max(degβ X(α, β), degβ Y (α, β)) = degβ F −1 (α, β), deg Q(X, Y ) ≤ max(degα X(α, β), degα Y (α, β)) = degα F −1 (α, β), deg F (X, Y ) = deg F −1 (α, β). Proof. The ﬁrst two inequalities were proved in (10) and (14) and the discussion that followed. The second two inequalities follow from the ﬁrst two by changing the roles of F and F −1 . The ﬁfth equality is a conclusion of the previous 4 inequalities. Remark 15. The ﬁfth equality appears in [2] on p. 292. It is proved there for any dimension (with the appropriate modiﬁcation). It was communicated to the authors of [2] by Ofer Gaber who attributed it to an unrecalled colloquium lecturer at Harvard. The authors mention that John Tyrrell (Kings College, Univ. of London) has indicated that this equality was well known to classical geometers. Theorem 15. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) (P (X, Y ), Q(X, Y )) is an invertible morphism. Let us denote F −1 (α, β) (X(α, β), Y (α, β)). Let the coeﬃcients of P (X, Y ) be {ai }, i.e., P (X, Y ) Z({ai })[X, Y ] and let the coeﬃcients of Q(X, Y ) be {bi }, i.e., Q(X, Y ) Z({bi })[X, Y ]. Then

= = ∈ ∈

X(α, β), Y (α, β) ∈ Q({ai }, {bi })[α, β]. Proof. This follows from Theorem 12 and the fact that R1 ∈ Z({ai }), S1 ∈ Z({bi }).

Corollary 8. Let P (X, Y ), Q(X, Y ) ∈ Q[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism, then F (Q2 ) = Q2 .

14 The Rigidity of Morphisms A remarkable property of invertible morphisms is the fact that the inverse map depends on very few of the coeﬃcients of the map. If the map is invertible and if we are given these very few coeﬃcients then we can reconstruct the map and hence there is only one morphism with such data. We call this property rigidity and we now describe it’s details. We shall need the following standard Deﬁnition 13. Let P (X, Y ) ∈ C[X, Y ]. The two face polynomials of P (X, Y ) are the following polynomials of a single indeterminate P (X, 0) and P (0, Y ). Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and let F (X, Y ) = (P (X, Y ), Q(X, Y )) be the corresponding polynomial map. The four face polynomials of F (X, Y ) are P (0, Y ), Q(0, Y ), P (X, 0), Q(X, 0).

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We shall also need the following notation Deﬁnition 14. Let f (T ), g(T ) ∈ C[T ]. The induced polynomial of f and g is deﬁned as follows Rf,g (α, β) = resultant(f (T ) − α, g(T ) − β), where α, β are two new indeterminates. Remark 16. Let f (T ), g(T ) ∈ C[T ] and let α, β be two indeterminates. (1) Clearly Rf,g (α, β) ∈ C[α, β]. (2) Let us assume that we have the following standard representations f (T ) = an T n + · · · + a0 , deg f (T ) = n, g(T ) = bm T m + · · · + b0 , deg g(T ) = m, then by Sylvester’s formula we have an ... 0 .. . ... an .. Rf,g (α, β) = a0 − α . . . . . .. .. ... . 0 . . . a0 − α

bm

...

.. .

...

b0 − β

...

.. .

...

0

...

bm .. . . .. . b0 − β 0

(3) Using the representation of Rf,g (α, β) in (2) above we obtain the following: degα Rf,g (α, β) = m = deg g(T ), and the α-leading term is + − bnm αm = + − (leading coeﬀ of g)deg f (T ) αdeg g(T ) , degβ Rf,g (α, β) = n = deg f (T ), and the β-leading term is n deg f (T ) am (leading coeﬀ of f )deg g(T ) β deg f (T ) , n (−β) = (−1) deg Rf,g (α, β) = max(n, m) = max(deg f (T ), deg g(T )).

We can now restate the previous results in terms of the rigidity property. Theorem 16 (The rigidity of morphisms). Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and let F (X, Y ) = (P (X, Y ), Q(X, Y )) be the corresponding polynomial map. Then we have the following:

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(a) If F (X, Y ) is an invertible morphism then the polynomial map G(α, β) = (RP (0,Y ),Q(0,Y ) (α, β), RP (X,0),Q(X,0) (α, β)), which is induced by the four face polynomials of F (X, Y ) is also an invertible morphism. (b) If F (X, Y ) is an invertible morphism then F −1 (α, β) diﬀers from G(α, β) by a dilation, i.e., there are R1 , S1 ∈ C∗ so that F −1 (α, β) = (−RP (0,Y ),Q(0,Y ) (α, β)/R1 , −RP (X,0),Q(X,0) (α, β)/S1 ). (c) If F (X, Y ) is an invertible morphism and G(α, β) = (X(α, β), Y (α, β)) then F (X, Y ) is an inner dilation of the polynomial map which is induced by the four face polynomials of G(α, β), i.e., there exist R1 , S1 ∈ C∗ so that F (X, Y ) = (RX(0,β),Y (0,β) (R1 X, S1 Y ), RX(α,0),Y (α,0) (R1 X, S1 Y )). Proof. All the three statements follow in a straight forward manner from Theorem 12. Remark 17. The core of the rigidity property lies in part (c) of the last theorem: for if we are given the four face polynomials of a morphism, namely, P1 (X) = P (X, 0), P2 (Y ) = P (0, Y ), Q1 (X) = Q(X, 0), Q2 (Y ) = Q(0, Y ) then the process (P2 (Y ), Q2 (Y )) → X(α, β) = RP2 ,Q2 (α, β) → (X(0, β), Y (0, β)) → P (R1 X, S1 Y ), and (P1 (Y ), Q1 (Y )) → Y (α, β) = RP1 ,Q1 (α, β) → (X(α, 0), Y (α, 0)) → Q(R1 X, S1 Y ), reconstructs morphism from it’s four face polynomials up to dilations. Note that the number of coeﬃcients of F (X, Y ) is

deg P (X, Y ) + 2 deg Q(X, Y ) + 2 + , 2 2 while the number of coeﬃcients of the four face polynomials is only deg P (X, 0) + deg P (0, Y ) + deg Q(X, 0) + deg Q(0, Y ) + 4, hence only a linear number (in the degrees) of the coeﬃcients out of the quadratic number of their total is used for the reconstruction of the morphism.

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It would have been nice if the rigidity property were a characterization of invertibility. Unfortunately, it is not. Example 8. We shall take a single polynomial P (X, Y ) ∈ C[X, Y ] and generate the map F (X, Y ) = (P (X, Y ), P (X, Y )) which is clearly not invertible. However, we shall see that for certain choices of P (X, Y ) the reconstruction process of Theorem 16(c) when applied to F (X, Y ) reconstructs F (X, Y ). Take P (X, Y ) = (X − Y )2 = Y 2 − 2XY + X 2 = X 2 − 2Y X + Y 2 . Then X(α, β) = RP (0,Y ),P (0,Y ) (α, β) = (α − β)2 = P (α, β). Similarly Y (α, β) = P (α, β). We note that in this case resultant((X − Y )2 − α, (X − Y )2 − β, Y ) = (α − β)2 , and (β − α)2 = (β − α)((X − Y )2 − α) − (β − α)((X − Y )2 − β). One might suspect that when the process of Theorem 16(c) is performed enough times then it might stabilize either on an invertible morphism or on the other extreme, a map whose coordinates are algebraically dependent. Again, unfortunately this is not the case. Example 9. Let P (X, Y ) = X 2 + Y 2 + X = Y 2 + (X 2 + X) = X 2 + X + Y 2 , Q(X, Y ) = XY + Y = (1 + X)Y. Then

and

1 RP (0,Y ),Q(0,Y ) (α, β) = 0 −α

0 1 = β 2 − α, −β

1 −β 0

1 RP (X,0),Q(X,0) (α, β) = 1 −α

0 −β 0

0 0 = β 2 . −β

So the next stage is to work with P1 (X, Y ) = Y 2 − X,

Q1 (X, Y ) = Y 2 ,

RP1 (0,Y ),Q1 (0,Y ) (α, β) = (α − β)2 , RP1 (X,0),Q1 (X,0) (α, β) = β 2 . So the next stage is to work with

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P2 (X, Y ) = (Y − X)2 = Y 2 − 2XY + X 2 ,

Q2 (X, Y ) = Y 2 ,

RP2 (0,Y ),Q2 (0,Y ) (α, β) = (α − β)2 , RP2 (X,0),Q2 (X,0) (α, β) = β 2 . So P3 (X, Y ) = P2 (X, Y ) and Q3 (X, Y ) = Q2 (X, Y ). The process had stabilized, however: ∂(P3 , Q3 )/∂(X, Y ) = 4Y (X − Y ) ≡ 0. Remark 18. The examples suggest that iterations of the process of Theorem 16(c) stabilize. If so, what are the possible periods? Theorem 12 gives an inversion formula in terms of the face polynomials. Theorem 13 gives an inversion formula in terms of the logarithmic derivatives of the two resultants at the origin. We now add one more inversion formula in terms of the polynomial coeﬃcients in the resultant representation, namely: Theorem 17. Let P (X, Y ), Q(X, Y )) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism normalized by F (0, 0) = (0, 0). Let resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = A(X, Y, α, β)(P (X, Y ) − α) + B(X, Y, α, β)(Q(X, Y ) − β), and resultant(P (X, Y ) − α, Q(X, Y ) − β, X) = A1 (X, Y, α, β)(P (X, Y ) − α) + B1 (X, Y, α, β)(Q(X, Y ) − β). Then F −1 (X, Y ) = ((XA(0, 0, X, Y ) + Y B(0, 0, X, Y ))/R1 , (XA1 (0, 0, X, Y ) + Y B1 (0, 0, X, Y ))/S1 ), for some R1 , S1 ∈ C∗ . Proof. The polynomial A(X, Y, α, β)(P (X, Y ) − α) + B(X, Y, α, β)(Q(X, Y ) − β) is independent of Y and so equals to A(X, 0, α, β)(P (X, 0) − α) + B(X, 0, α, β)(Q(X, 0) − β). To obtain the X-free term R0 (α, β) of resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) we substitute in this polynomial X = 0 and obtain R0 (α, β) = A(0, 0, α, β)(P (0, 0) − α) + B(0, 0, α, β)(Q(0, 0) − β) = −αA(0, 0, α, β) − βB(0, 0, α, β). We do the same for the second resultant and obtain αA1 (0, 0, α, β) − βB1 (0, 0, α, β), and now the result follows by Theorem 12.

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15 The Fibre Theorem The following result is well known. Theorem (The Fibre Theorem). If k is an algebraically closed ﬁeld, F : k n → k n a polynomial map such that det J(F ) never vanishes, then the cardinality of the ﬁbre F −1 ({b}) over any b ∈ k n is less than or equal to the geometric degree of F , dimk(F ) k(X) which is ﬁnite. See [12]. This was generalized by Arno van den Essen to ﬁelds k of characteristic 0 [6]. For the ﬁeld k = R and in dimension n = 2 the ﬁniteness of the ﬁbres was proved in 1988 [14]. See also [1, 2]. This problem was handled in a geometrical fashion for maps that are not even polynomial. See [18, 19]. The purpose of this small section is to give one more proof for the ﬁniteness result of the ﬁbres of polynomial ´etale maps in dimension two and over an algebraically closed ﬁeld, using the ideas that were developed in the previous sections. Theorem 18 (The Fibre Theorem). If F : C2 → C2 is a polynomial map such that det J(F ) never vanishes, then for each b ∈ C2 we have F −1 ({b}) ≤ (degX F )(degY F ). Proof. Let F (X, Y ) = (P (X, Y ), Q(X, Y )) where P (X, Y ), Q(X, Y ) ∈ C[X, Y ]. Let α, β be two new indeterminates. We use our standard notation resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = RN (α, β)X N + · · · + R0 (α, β), where we know (by our previous results) that N ≥ 1 and that Rj (α, β) ∈ C[α, β] for 0 ≤ j ≤ N . There exist A(X, Y, α, β), B(X, Y, α, β) ∈ C[X, Y, α, β] such that the following holds true RN (α, β)X N + · · · + R0 (α, β) = A(X, Y, α, β)(P (X, Y ) − α) + B(X, Y, α, β)(Q(X, Y ) − β). For any b = (α0 , β0 ) ∈ C2 the X-ﬁbre of F (X, Y ) = b equals the X-zero set of RN (α0 , β0 )X N + · · · + R0 (α0 , β0 ). This set contains at most N points. Let us recall once more that by the Sylvester’s formula for resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) we have N ≤ max(degY P (X, Y ), degY Q(X, Y )) = degY F (X, Y ). Similarly the set of Y -ﬁbre of F (X, Y ) = b contains at most degX F (X, Y ) points. Hence we obtain F −1 ({b}) ≤ (degX F )(degY F ).

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16 One more Inversion Formula and an Equivalent Formulation to the Jacobian Conjecture We start by recalling well known facts about resultants: Let f (X), g(X) ∈ C[X]. Let us assume that we have the following standard representations: f (X) = am X m + · · · + a0 = am

m

(X − αi ),

i=1

and g(X) = bn X n + · · · + b0 = bn

n

(X − βj ).

j=1

Then the following holds true: resultant(f, g, X) = anm

m

g(αi )

i=1

= (−1)mn bm n

n j=1

f (βj ) = anm bm n

m n

(αi − βj ).

i=1 j=1

Suppose now that P (X, Y ), Q(X, Y ) ∈ C[X, Y ] is a Jacobian pair, i.e., ∂(P, Q)/∂(X, Y ) = PX QY − PY QX ≡ 1. Let us use the following assumptions and notations: n

P (X, Y ) − α = an Y n + · · · + a0 − α = an Q(X, Y ) − β = bn Y n + · · · + b0 − β = bn

(Y − αi ),

i=1 n

(Y − βj ),

j=1

where ai , bj ∈ C[X], 0 ≤ i, j ≤ n, an , bn ∈ C∗ , αi = αi (X, α) are algebraic over C[X, α], βj = βj (X, β) are algebraic over C[X, β]. ∂/∂X{P (X, Y ) − α} = −an × ∂/∂Y {P (X, Y ) − α} = an

n

i=1 αi (Y n

(Y − α1 ) · · · (Y − αi−1 ) − αi+1 ) . . . (Y − αn ),

(Y − α1 ) · · · (Y − αi−1 )(Y − αi+1 ) · · · (Y − αn ),

i=1

The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which

∂/∂X{Q(X, Y ) − β} = −bn

n

j=1 × βj (Y n

193

(Y − β1 ) · · · (Y − βj−1 ) − βj+1 ) · · · (Y − βn ),

(Y − β1 ) · · · (Y − βj−1 )(Y − βj+1 ) · · · (Y − βn ).

∂/∂Y {Q(X, Y ) − β} = bn

j=1

Hence PX (X, αi ) = −an (αi − α1 ) · · · (αi − αi−1 )αi (αi − αi+1 ) · · · (αi − αn ), and PY (X, αi ) = an (αi − α1 ) · · · (αi − αi−1 )(αi − αi+1 ) · · · (αi − αn ). Proposition 4. 1 = −PY (X, αi )d/dX{Q(X, αi )} = QY (X, βj )d/dX{P (X, βj )},

1 ≤ i, j ≤ n.

Proof. 1 = PX (X, αi )QY (X, αi ) − PY (X, αi )QX (X, αi ) = [−an (αi − α1 ) · · · (αi − αi−1 )αi (αi − αi+1 ) · · · (αi − αn )]QY (X, αi ) − [an (αi − α1 ) · · · (αi − αi−1 )(αi − αi+1 ) · · · (αi − αn )]QX (X, αi ) = −[an (αi − α1 ) · · · (αi − αi−1 )(αi − αi+1 ) · · · (αi − αn )]{QX (X, αi ) + αi QY (X, αi )} = −PY (X, αi )d/dX{Q(X, αi )}. A similar computation gives 1 = QY (X, βj )d/dX{P (X, βj )}.

Theorem 19. Let R(X) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ). Then −R (X)/R(X) =

n

{1/PY (X, αi )[Q(X, αi ) − β]}

i=1

=−

n

{1/QY (X, βj )[P (X, βj ) − α]}.

i=1

Proof. By the previous proposition we have d/dX{Q(X, αi )−β}/{Q(X, αi )−β} = −1/PY (X, αi )[Q(X, αi )−β], Hence

1 ≤ i ≤ n.

194

Ronen Peretz n

d/dX{Q(X, αi ) − β}/{Q(X, αi ) − β}

i=1

=−

n

1/PY (X, αi )[Q(X, αi ) − β],

i=1

or

n

ann

d/dX

(Q(X, αi ) − β)

i=1

=−

n

ann

n

(Q(X, αi ) − β)

i=1

1/PY (X, αi )[Q(X, αi ) − β].

i=1

So −R (X)/R(X) =

n {1/PY (X, αi )[Q(X, αi ) − β]}, i=1

where in the last step we used the identity resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ) = ann

n

(Q(X, αi ) − β).

i=1

A similar computation gives the following: d/dX{P (X, βj ) − α}/{P (X, βj ) − α} = 1/QY (X, βj )[P (X, βj ) − α], 1 ≤ j ≤ n, n

d/dX{P (X, βj ) − α}/{P (X, βj ) − α}

j=1

=

1/QY (X, βj )[P (X, βj ) − α],

j=1

d/dX

n

2 (−1)n bnn

n

(P (X, βj ) − α)

j=1

=

n

2 (−1)n bnn

n

(P (X, βj ) − α)

j=1

1/QY (X, βj )[P (X, βj ) − α],

j=1

R (X)/R(X) =

n

1/QY (X, βj )[P (X, βj ) − α].

j=1

Theorem 20. Let P (X, Y ), Q(X, Y ) ∈ C[X, Y ] and suppose that F (X, Y ) = (P (X, Y ), Q(X, Y )) is an invertible morphism. Let us denote F −1 (α, β) = (X(α, β), Y (α, β)). Then we have the following formulas for the inverse map

The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which

X(α, β) = =

n

−1 J/PY (0, αi (0, α))[Q(0, αi (0, α)) − β]

i=1

−

195

n

−1 J/QY (0, βj (0, β))[P (0, βj (0, β)) − α]

,

j=1

where J = ∂(P, Q)/∂(X, Y ) ∈ C∗ . An analogous pair of expressions hold for Y (α, β). Proof. By Theorem 13 we have X(α, β) = −R(X)/d/dX{R(X)}(0), where R(X) = resultant(P (X, Y ) − α, Q(X, Y ) − β, Y ), and now the result follows by Theorem 19 above with J replacing 1 in the numerators. Theorem 21. The Jacobian conjecture is true iﬀ the following holds true: If P (X, Y ), Q(X, Y ) ∈ C[X, Y ] satisfy following two conditions (1) ∂(P, Q)/∂(X, Y ) ≡ 1 and n n (2) P (X, Y ) = an i=1 (Y − αi (X)), Q(X, Y ) = bn j=1 (Y − βj (X)), where a n , b n ∈ C∗ then n n Q(X, αk (X)) PY (X, αi (X))Q(X, αi (X)) ∈ C∗ . i=1

k=1

Moreover, this condition is equivalent to each of the following 3 conditions n n P (X, βk (X)) PY (X, αi (X))Q(X, αi (X)) ∈ C∗ , i=1

k=1

n

n

j=1

and n j=1

Q(X, αk (X))

QY (X, βj (X))P (X, βj (X))

∈ C∗ ,

k=1

n

P (X, βk (X))

QY (X, βj (X))P (X, βj (X))

∈ C∗ .

k=1

Proof. The Jacobian conjecture is true ⇐⇒ If P, Q ∈ C[X, Y ] satisfy ∂(P, Q)/∂(X, Y ) ≡ 1 then F = (P, Q) is an invertible morphism ⇐⇒

196

Ronen Peretz

If P, Q ∈ C[X, Y ] satisfy ∂(P, Q)/∂(X, Y ) ≡ 1 then R(X) = resultant(P − α, Q − β, Y ) = R1 X + R0 (α, β), where R1 ∈ C∗ (by Theorem 11 and Remark 12) ⇐⇒ If P, Q ∈ C[X, Y ] satisfy ∂(P, Q)/∂(X, Y ) ≡ 1 then R (X) ∈ C∗ ⇐⇒ If P, Q ∈ C[X, Y ] satisfy ∂(P, Q)/∂(X, Y ) ≡ 1 then R(X) ×

n

1/PY (X, αi )[Q(X, αi ) − β] ∈ C∗ .

i=1

Equivalently (by Theorem 19 above) R(X) ×

n

1/QY (X, βj )[P (X, βj ) − α] ∈ C∗ .

j=1

However, for any (α, β), and any P, Q ∈ C[X, Y ] we have: (P, Q) is a Jacobian pair ⇐⇒ (P − α, Q − β) is a Jacobian pair Hence we may take above n n α = β = 0. Finally, R(X) equals k=1 P (X, βk ) and also k=1 Q(X, αk ) up to a C∗ factor. This completes the proof. Theorem 22. The Jacobian conjecture is true iﬀ the following holds true: If P (X, Y ), Q(X, Y ) ∈ C[X, Y ] satisfy the following two conditions (1) ∂(P, Q)/∂(X, Y ) ≡ 1 and n n (2) P (X, Y ) = an i=1 (Y − αi (X)), Q(X, Y ) = bn j=1 (Y − βj (X)), where a n , b n ∈ C∗ then n lim X/PY (X, αi )Q(X, αi ) = −1. X→∞

i=1

Moreover, this condition is equivalent to the following condition lim

X→∞

n

X/QY (X, βj )P (X, βj ) = −1.

j=1

Proof. By Theorem 19 above we have −R (X)/R(X) = =

n i=1 n j=1

1/PY (X, αi )Q(X, αi ) 1/QY (X, βj )P (X, βj ),

The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which

197

where we picked α = β = 0 (we use for that the same argument that was used in the proof of the previous theorem). If R(X) = Rl X l + · · · + R0 , then

Rl = 0,

lim XR (X)/R(X) = l.

X→∞

Hence R(X) = R1 X + R0 , R1 = 0 iﬀ limX→∞ XR (X)/R(X) = 1 and the proof is completed as the proof of the previous theorem. n n Theorem 23. Let P (X, Y ) = an i=1 (Y − αi (X)), Q(X, Y ) = bn j=1 (Y − βj (X)) ∈ C[X, Y ]. Let R(X) = resultant(P (X, Y ), Q(X, Y ), Y ). Then the following are true: (a) If R(X0 ) = 0 then there are 1 ≤ i0 , j0 ≤ n so that αi0 (X0 ) = βj0 (X0 ). (b) If P (X, Y ) is regular and if PY (X0 , αi0 (X0 )) = 0 then lim |αi 0 (X)| = ∞.

X→X0

(c) If (P, Q) is a Jacobian pair and if PY (X0 , αi0 (X0 )) = 0 but R(X0 ) = 0 then there exists a j0 = i0 so that PY (X0 , αj0 (X0 )) = 0. Proof. (a) We have R(X) = A(X, Y )P (X, Y ) + B(X, Y )Q(X, Y ) and as is well known, if R(X0 ) = 0 then there exists a Y0 so that P (X0 , Y0 ) = Q(X0 , Y0 ) = 0. Thus αi0 (X0 ) = βj0 (X0 ) for some 1 ≤ i0 , j0 ≤ n. (b) For an open Zariski set αi 0 (X) exists. By P (X, αi0 (X)) ≡ 0 we obtain on this open set PX (X, αi0 (X)) + αi 0 (X)PY (X, αi0 (X)) ≡ 0. Letting X → X0 we have: lim PY (X, αi0 (X)) = PY (X0 , αi0 (X0 )) = 0.

X→X0

So by regularity lim PX (X, αi0 (X)) = PX (X0 , αi0 (X0 )) = 0.

X→X0

But then by lim {PX (X, αi0 (X)) + αi 0 (X)PY (X, αi0 (X))} = 0,

X→X0

we deduce that

lim |αi 0 (X)| = ∞,

X→X0

as desired.

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Ronen Peretz

(c) In this case, by Theorem 19 above we have R (X)/R(X) = −

n

1/PY (X, αi (X))Q(X, αi (X)).

i=1

Since R(X0 ) = 0 it follows that lim

X→X0

n

1/PY (X, αi (X))Q(X, αi (X)) = −R (X0 )/R(X0 ) ∈ C.

i=1

It also follows by R(X0 ) = 0 that for each i = 1, . . . , n we have Q(X0 , αi (X0 )) = 0. Since PY (X0 , αi0 (X0 )) = 0 it follows from the above that there must be some cancellation which must be of the form PY (X0 , αj0 (X0 )) = 0 for some j0 = i0 . This concludes the proof of part (c).

17 Parametrization of the Jacobian Variety In this section we address in a very elementary manner the problem of parametrizing the Jacobian variety. If such a parametrization is available it will usually carry with it some useful information about the variety. For example it will tell us it’s dimension. − 1) coeﬃcients for two polynomials of degree In general we have 2( n+2 2 n (at most) and the Jacobian variety is determined by 2n 2 equations on these coeﬃcients. All the equations are quadratic-expressible linearly in terms of the Grassmann coordinates of the line that passes through the two points that are determined by the coeﬃcients of the two polynomials. All the equations except for one are homogeneous. Let us denote: aij X i Y j , P (X, Y ) = 1≤i+j≤n

Q(X, Y ) =

bij X i Y j .

1≤i+j≤n

Then ∂P/∂X =

iaij X i−1 Y j ,

∂P/∂Y =

1≤i+j≤n

∂Q/∂X =

1≤i+j≤n

So

jaij X i Y j−1 ,

1≤i+j≤n

ibij X

i−1

j

Y ,

∂Q/∂Y =

1≤i+j≤n

jbij X i Y j−1 .

The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which

∂(P, Q)/∂(X, Y ) =

iaij X i−1 Y j

1≤i+j≤n

−

1≤i+j≤n

=

jbij X i Y j−1

1≤i+j≤n

jaij X i Y j−1

199

ibij X i−1 Y j

1≤i+j≤n

i1 j2 (ai1 j1 bi2 j2 − ai2 j2 bi1 j1 )X i1 +i2 −1 Y j1 +j2 −1 .

1≤i1 +j1 ≤n 1≤i2 +j2 ≤n

We note that if we put i = i1 + i2 − 1, j = j1 + j2 − 1 then i2 = i − i1 + 1, j2 = j − j1 + 1. Also 1 ≤ i1 + j1 ≤ n and 1 ≤ i2 + j2 ≤ n. The last double inequality is 1 ≤ (i − i1 + 1) + (j − j1 + 1) ≤ n, or i + j + 2 − n ≤ i1 + j1 ≤ i + j + 1. Combining that with 1 ≤ i1 + j1 ≤ n we obtain max(1, i + j + 2 − n) ≤ i1 + j1 ≤ min(n, i + j + 1). This proves the following. Theorem 24. The Jacobian variety of degree n is given by the following equations: i1 (j − j1 + 1)

2n 2

max(1,i+j+2−n)≤i1 +j1 ≤min(n,i+j+1)

× [ai1 j1 b(i−i1 +1)(j−j1 +1) − a(i−i1 +1)(j−j1 +1) bi1 j1 ] = 0 a10 b01 − a01 b10 = J, where 1 ≤ i + j ≤ 2(n − 1). Deﬁnition 15. Let us arrange the coeﬃcients {bij }0≤i+j≤n in a certain orequations in der, say, the lexicographical order. Then we may view the 2n 2 Theorem 24 as a linear system which is non-homogeneous, of p = 2n equa2 n+2 tions in the q = 2 − 1 unknowns {bij }0≤i+j≤n . The matrix of the coeﬃcients of the system has for it’s entries integral multiples of certain {aij } such that the sum of the coeﬃcients in each row is 0. We shall denote this matrix that depends only on P (X, Y ) by MP . We can now give a linear algebraic characterization of polynomials that have Jacobian mates. Theorem ⎞ Let P (X, Y ) ∈ C[X, Y ] and denote n = deg P (X, Y ) and ⎛ 25. 0 . e(2n) = ⎝ .. ⎠ the unit vector of dimensions 2n 2 × 1. P (X, Y ) has a Jacobian 2 0 1

mate iﬀ

200

Ronen Peretz

rank {MP } = rank {MP e(2n) }. 2

Here MP e(2n) is the matrix MP augmented by the column e(2n) . 2 2 Proof. The following fact is a well-known, see Theorem X on p. 85 of [10]: A necessary and suﬃcient condition that the non-homogeneous equations q

αij Xj = Ci

(i = 1, . . . , p),

j=1

have a solution is that the matrices ⎛ ⎞ a11 . . . a1q ⎜ ⎟ ⎜ .. .. .. ⎟ and ⎜ . . . ⎟ ⎝ ⎠ ap1

...

apq

⎛

a11

⎜ ⎜ .. ⎜ . ⎝ ap1

...

a1q

.. .

.. .

...

apq

C1

⎞

⎟ .. ⎟ , . ⎟ ⎠ Cp

have the same rank. Clearly P (X, Y ) has a Jacobian mate iﬀ the system of equations of Theorem 24 has a solution {bij }, 1 ≤ i + j ≤ n. However, this is a linear system in the {bij } and it’s coeﬃcient matrix is MP while it’s column of free elements is Je(2n) . This proves the assertion. 2

As noted above and as is clear from Theorem 24 the Jacobian variety is determined by 2n 2 − 1 homogeneous equations plus one more non-homogeneous equation, namely, a10 b01 − a01 b10 = J. With the lexicographical order this last equation corresponds to the last row of MP . This row is: (0, . . . , 0, −a01 , a10 ). n+2 − Thus MP has the following structure: it consists of an ( 2n 2 −1) by an ( 2 2n 1) matrix HP that corresponds to the 2 − 1 homogeneous equations, sitting on the top of the row (0, . . . , 0, −a01 , a10 ). n+2 − 1) matrix HP is called the homoDeﬁnition 16. The ( 2n 2 − 1) × ( 2 geneous matrix of the polynomial P (X, Y ). We can now give a second linear algebraic characterization of polynomials that have Jacobian mates. Theorem 26. Let P (X, Y ) ∈ C[X, Y ]. P (X, Y ) has a Jacobian mate iﬀ rank {MP } = 1 + rank {HP }. Proof. Let n = deg P (X, Y ) and r = rank {MP }. By Theorem 25 P (X, Y ) has a Jacobian mate iﬀ rank {MP } = rank {MP e(2n) }. 2

The Asymptotic Values of a Polynomial Map C2 → C2 form a Variety Which

201

Since r = rank {MP } it follows that MP contains an r × r submatrix which is regular and that any (r + 1) × (r + 1) submatrix of MP is singular. We claim that the condition rank {MP } = rank {MP e(2n) }, 2

is equivalent to the following condition: Any r × r regular submatrix of MP must contain entries in the last row of MP , namely, in (0, . . . , 0, −a01 , a10 ). For if there was an r × r regular submatrix of MP containing no entries in the last row then this submatrix had also to be a submatrix of HP and so clearly MP e(2n) had to contain a regular (r + 1) × (r + 1) submatrix (one of 2 n+1 whose entries is the 1 in ( 2n 2 , 2 )). So that rank {MP e(2n) } ≥ r + 1, 2

which is not possible. Hence HP contains no submatrix of order r×r which is regular but contains regular submatrices of order (r − 1) × (r − 1). Hence rank {HP } = r − 1 = rank {MP } − 1.

Acknowledgments The author is pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria for partial support of this project. The author would also like to thank the anonymous referees for helpful criticism as well as the editor of these proceedings, Misha Klin for the wonderful editorial job he did on this paper.

References 1. K. Adjamagbo, H. Derksen, and A. Van den Essen, On Polynomial Maps in Positive Characteristic and the Jacobian Conjecture, Report 9208, Univ. of Nijmegen, 1992 2. H. Bass, E. Connell, and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (2), 7 (1982), 287–330. 3. B. Buchberger, Gr¨ obner Bases: an algorithmic method in polynomial ideal theory, in N. K. Bose (ed.) Multidimensional Systems Theory, pp. 184–232, Reidel, Dordrecht, 1985.

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4. T. W. Dube, The Structure of Polynomial Ideals and Gr¨ obner Bases, Preprint, 1990 5. A. van den Essen, A note on Meisters and Olech’s proof of the global asymptotic stability Jacobian conjecture, Pac. J. Math. (2), 151 (1991), 351–357. 6. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, Vol. 190, Birkh¨ auser, Basel, 2000. 7. C. Gutierrez, and N. Van Chau, Properness and the Jacobian conjecture in R2 , Vietnam J. Math. (4), 31 (2003), 421–427. 8. R. Gebauer and H. M. Moller, On an installation of Buchberger’s algorithm, in L. Robbiano (ed.) Computational Aspects of Commutative Algebra, pp. 141–152, Academic Press, New York, 1988. 9. J. Hadamard, Sur les transformations pontuelles, Bull. de la Soc. Math. de France, 34 (1906), 71–81. 10. W. V. D. Hodge, and D. Pedoe, Methods of Algebraic Geometry, Vol. 1, Cambridge University Press, Cambridge, 1947. 11. O. H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys., 47 (1939), 299–306. 12. K. Kurdyka and K. Rusek, Surjectivity of certain injective semialgebraic transformations of Rn , Math. Z., 200 (1988), 141–148. 13. E. Mayr and A. Meyer, The complexity of the word problem for commutative semigroups and polynomial ideals, Adv. Math., 46 (1982), 305–329. 14. G. Meisters and C. Olech, Solution of the global asymptotic stability Jacobian conjecture for the polynomial case, in Analyse Mathematique et applications, pp. 373–381, Gautier-Villars, Paris, 1998. 15. T. Moh, On the global Jacobian Conjecture and the conﬁguration of roots, J. Reine Angew. Math., 340 (1983), 140–212. 16. H. M. Moller and F. Mora, Upper and lower bounds for the degree of Gr¨ obner bases, in EUROSAM 1984, Lecture Notes in Computer Science, Vol. 174, pp. 172–183, Springer, Berlin, 1984. 17. P. Nousiainen and M. E. Sweedler, Automorphisms of polynomial and power series rings, J. Pure Appl. Algebra, 29 (1983), 93–97. 18. R. Peretz, The Topology of Maximal Domains for Local Homeomorphism Mappings on R2 and an Application to the Jacobian Conjecture, Technion Preprint Series NO-MT, 833, 1988. 19. R. Peretz, Maximal domains for entire functions, J. Anal. Math., 61 (1993), 1–28. 20. R. Peretz, The variety of the asymptotic values of a real polynomi al `etale map, J. Pure Appl. Algebra, 01/106 (1996), 103–112. 21. R. Peretz, On counterexamples to Keller’s problem, Illinois J. Math. (02), 40 (1996), 293–303. 22. R. Peretz, The geometry of the asymptotics of polynomial maps, Israel J. Math., 105 (1998), 1–59. 23. S. Pinchuk, A counterexample to the strong real Jacobian conjecture, Math. Z., 217 (1994), 1–4.

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24. M. Razar, Polynomial maps with constant Jacobian, Israel J. Math., 32 (1979), 97–106. 25. N. Van Chau, Two remarks on non-zero constant Jacobian polynomial maps of C2 , Ann. Pol. Math. (1), 82 (2003), 39–44. 26. D. Wright, On the Jacobian conjecture, Illinois J. Math., 25 (1981), 423– 440.

Part B

Research Papers

Some Meeting Points of Gr¨ obner Bases and Combinatorics B´alint Felszeghy1,2 and Lajos R´onyai1,2 1

2

Computer and Automation Institute, Hungarian Academy of Science, Budapest, Hungary. [email protected] Institute of Mathematics, Budapest University of Technology and Economics, Budapest, Hungary. [email protected]

Summary. Let F be a ﬁeld, V ⊆ Fn be a set of points, and denote by I(V ) the vanishing ideal of V in the polynomial ring F[x1 , . . . , xn ]. Several interesting algebraic and combinatorial problems can be formulated in terms of some ﬁnite V , and then Gr¨ obner bases and standard monomials of I(V ) yield a powerful tool for solving them. We present the Lex Game method, which allows one to eﬃciently compute the lexicographic standard monomials of I(V ) for any ﬁnite set V ⊆ Fn . We apply this method to determine the Gr¨ obner basis of I(V ) for some V of combinatorial and algebraic interest, and present four applications of this type. We give a new easy proof of a theorem of Garsia on a generalization of the fundamental theorem of symmetric polynomials. We also reprove Wilson’s theorem concerning the modulo p rank of some inclusion matrices. By examining the Gr¨ obner basis of the vanishing ideal of characteristic vectors of some speciﬁc set systems, we obtain results in extremal combinatorics. Finally, we point out a connection among the standard monomials of I(V ) and I(V c ), where V ⊆ {0, 1}n and V c = {0, 1}n \ V . This has immediate consequences in combinatorial complexity theory. The main results have appeared elsewhere in several papers. We collected them into a uniﬁed account to demonstrate the usefulness of Gr¨ obner basis methods in combinatorial settings.

Key words: Gr¨ obner basis, Standard monomial, Lexicographic order, Vanishing ideal, Hilbert function, Inclusion matrix, Rank formula

1 Introduction Throughout the paper n will be a positive integer, and [n] stands for the set {1, 2, . . . , n}. The set of all subsets of [n] is denoted by 2[n] . Subsets of 2[n] are called set families or set systems. Let [n] m denotethefamily of all m[n] subsets of [n] (subsets which have cardinality m), and ≤m is the family of those subsets that have at most m elements. By N we mean the nonnegative M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 6,

207

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integers, Z is the set of integers, Q is the ﬁeld of rational numbers, and Fp is the ﬁeld of p elements, where p is a prime. Let F be a ﬁeld. As usual, we denote by F[x1 , . . . , xn ] = F[x] the ring of polynomials in variables x1 , . . . , xn over F. To shorten our notation, we write f (x) for f (x1 , . . . , xn ). Vectors of length n are denoted by boldface letters, for wn 1 we write xw for xw example y = (y1 , . . . , yn ) ∈ Fn . If w ∈ Nn , 1 . . . xn ∈ F[x]. For a subset M ⊆ [n], the monomial xM is i∈M xi (and x∅ = 1). We say that a polynomial is multilinear if it is a linear combination of some xM (M ⊆ [n]). Suppose that V ⊆ Fn . Then the vanishing ideal I(V ) of V consists of polynomials in F[x], which as functions vanish on V . In our applications, we consider ﬁnite sets V , and use the Gr¨obner bases, or standard monomials of I(V ) (see the next subsection for the deﬁnitions) to prove claims on V . Let vF ∈ {0, 1}n denote the characteristic vector of a set F ⊆ [n], that is the ith coordinate of vF is 1 iﬀ i ∈ F . For a system of sets F ⊆ 2[n] , let us put VF for the set of the characteristic vectors of elements of F . By I(F ) we understand the vanishing ideal I(VF ), as it will make no confusion. In Sect. 2 we collected the deﬁnitions and basic facts we need about Gr¨ obner bases and Hilbert functions. We develop a combinatorial description of the lexicographic standard monomials of I(V ) in the subsequent section via a two player game. Lea and Stan play the Lex Game with some ﬁxed parameters V ⊆ Fn and w ∈ Nn . We show that xw is a lexicographic standard monomial of I(V ) if and only if Stan has a winning strategy in the game. This description proves to be more than just a toy. It yields a fast algorithm to determine the standard monomials of I(V ) for an arbitrary ﬁnite V . On the other hand, it is also applicable in the ’symbolic’ computation of the standard monomials for some particular sets V . We shall see several examples of such calculations in Sect. 4, which is devoted to combinatorial and algebraic applications. We give a new easy proof of a theorem of Garsia on a generalization of the fundamental theorem of symmetric polynomials. We also reprove Wilson’s theorem concerning the modulo p rank of some inclusion matrices. In the direction of extremal combinatorics, we obtain results on the maximal cardinality of some set systems. To be a bit more speciﬁc, we will consider modulo q L-avoiding L-intersecting families, and families that do not shatter large sets. The last application is to point out a connection among the standard monomials and Hilbert functions of I(V ) and I(V c ), where V ⊆ {0, 1}n and V c = {0, 1}n \ V . An immediate consequence of this in combinatorial complexity theory is shown. Much of the results described here have already appeared elsewhere, most notably in [12, 19, 16, 11, 15, 22]. In some cases the way of exposition, which is based primarily on the Lex Game, is new and considerably simpler than the original one. We collected the material to point out interesting combinatorial applications of Gr¨ obner basis methods.

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2 Preliminaries 2.1 Gr¨ obner Bases and Standard Monomials We recall now some basic facts concerning Gr¨obner bases in polynomial rings over ﬁelds. More detailed exposition can be found in the classic papers by prof. Bruno Buchberger [3–5], and in the textbook [9]. A total order ≺ on the monomials composed from variables x1 , x2 , . . . , xn is a term order, if 1 is the minimal element of ≺, and ≺ is compatible with multiplication with monomials. Two important term orders are the lexicographic (lex for short) and the degree compatible lexicographic (deglex ) orders. We have xw ≺lex xu if and only if wi < ui holds for the smallest index i such that wi = ui . As for deglex, we have that a monomial of smaller degree is smaller in deglex, and among monomials of the same degree lex decides the order. Also in general, ≺ is degree compatible, if deg(xw ) < deg(xu ) implies xw ≺ xu . The leading monomial (or leading term) lm(f ) of a nonzero polynomial f ∈ F[x] is the largest monomial (with respect to ≺) which appears with nonzero coeﬃcient in f , when written as the usual linear combination of monomials. It is easy to verify that the leading monomial of a product f · g of nonzero polynomials is lm(f ) · lm(g). We denote the set of all leading monomials of polynomials of a given ideal I F[x] by Lm(I) = {lm(f ) : f ∈ I}, and we simply call them the leading monomials of I. A monomial is called a standard monomial of I, if it is not a leading monomial of any f ∈ I. Let Sm(I) denote the set of standard monomials of I. Obviously, Sm(I) is a downset with respect to division, that is, a divisor of a standard monomial is again in Sm(I). A ﬁnite subset G ⊆ I is a Gr¨ obner basis of I, if for every f ∈ I there exists a g ∈ G such that lm(g) divides lm(f ). Using that ≺ is a well founded order, it follows that G is actually a basis of I, that is, G generates I as an ideal of F[x]. It is a fundamental fact that every nonzero ideal I of F[x] has a Gr¨obner basis. A Gr¨ obner basis G ⊆ I is reduced if for all g ∈ G, the leading coeﬃcient of g (i.e. the coeﬃcient of lm(g)) is 1, and g = h ∈ G implies that no nonzero monomial in g is divisible by lm(h). This is clearly equivalent to saying that every g ∈ G has leading coeﬃcient 1, {lm(g) : g ∈ G} is the set of minimal elements of Lm(I) (with respect to division), and the polynomial g − lm(g) is a linear combination of standard monomials. For any ﬁxed term order and any nonzero ideal of F[x] there exists a unique reduced Gr¨ obner basis. Suppose that f ∈ F[x] contains a monomial xw · lm(g), where g is some other polynomial with leading coeﬃcient c. Then we can reduce f with g (and get fˆ), that is, we can replace xw · lm(g) in f with xw · (lm(g) − 1c g). Clearly if g ∈ I, then f and fˆ represent the same coset in F[x]/I. Also note that lm(xw · (lm(g) − 1c g)) ≺ xw · lm(g). As ≺ is a well founded order, this guarantees that if we reduce f repeatedly with a set of polynomials G, then

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we end up with a reduced fˆ in ﬁnitely many steps, that is a polynomial such that none of its monomials is divisible by any lm(g) (g ∈ G). Assume now that G is a Gr¨obner basis of some ideal I. In this case, it can be shown that the reduction of any polynomial with G is unique. We see from the deﬁnitions that the reduction fˆ of a polynomial f is a linear combination of standard monomials of I. From these, it follows directly that for a nonzero ideal I the set Sm(I) is a linear basis of the F-vectorspace F[x]/I. If I(V ) is a vanishing ideal of a ﬁnite set V of points in Fn , then F[x]/I(V ) can be interpreted as the space of functions V → F. An immediate consequence is that the number of standard monomials of I(V ) is |V |. In particular for every family of sets we have |F | = |Sm(I(F ))|. Another property of the standard monomials of I(F ) will be needed several times: for an arbitrary set family F , one has x2i − xi ∈ I(F ), therefore all the elements of Sm(I(F )) are multilinear monomials. 2.2 The Hilbert Function We write F[x]≤m for the vector space of polynomials over F with degree at most m. Similarly, if I F[x] is an ideal then I≤m = I ∩ F[x]≤m stands for the linear subspace of polynomials from I with degree at most m. The Hilbert function of the F-algebra F[x]/I is HI : N → N, where HI (m) = dimF F [x]≤m I≤m . Let ≺ be any degree compatible term ordering (deglex for instance). One can easily see that the set of standard monomials with respect to ≺ of degree at most m forms a linear basis of F[x]≤m I≤m . Hence we can obtain HI (m) by determining the set Sm(I) with respect to any degree compatible term ordering. When F is a system of sets, we call HI(F ) (m) the Hilbert function of F and denote it by HF (m), as it makes no confusion. In the combinatorial literature HF (m) is usually given in terms of inclusion matrices. For two families F , G ⊆ 2[n] the inclusion matrix I(F , G) is a matrix of size |F | × |G|, whose rows and columns are indexed by the elements of F and G, respectively. The entry at position (F, G) is 1 if G ⊆ F and 0 otherwise (F ∈ F , G ∈ G). It is a simple matter to verify that the Hilbert function of F is given by [n] . (1) HF (m) = dimF F [x]≤m I(F )≤m = rankF I F , ≤m We will beneﬁt from a similar statement in Sect. 4.2, which claims that [n] , dimF (PF ,m ) = rankF I F , m

(2)

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where PF ,m is the linear space of functions from VF to F which can be represented as homogeneous multilinear polynomials of degree m. (With a slight abuse of notation we could have written PF ,m = F[x]=m I(F )=m .) Incidence matrices and their ranks are important in the study of ﬁnite geometries as well. Standard monomials and Hilbert functions are also useful in that setting. The reader is referred to Moorhouse [20] in the present volume for an account on applications of this type.

3 Computation of the Lex Standard Monomials In this section we sketch a purely combinatorial description of the lexicographic standard monomials of vanishing ideals of ﬁnite sets of points. This is the main tool which can be applied to compute lex standard monomials of sets of combinatorial interest. The original source is [12], and the interested reader can ﬁnd an extension to general zero dimensional ideals in [13]. Throughout the section, we use the lexicographic ordering, so – even if it is not stated explicitly – Sm(I) and Lm(I) is deﬁned with respect to lex. As before, let F be a ﬁeld, V ⊆ Fn a ﬁnite set and w = (w1 , . . . , wn ) ∈ Nn an n dimensional vector of natural numbers. With these data ﬁxed, we deﬁne the Lex Game Lex(V ; w), which is played by two persons, Lea and Stan. Both Lea and Stan know V and w. 1 Lea chooses wn elements of F. Stan picks a value yn ∈ F, diﬀerent from Lea’s choices. 2 Lea now chooses wn−1 elements of F. Stan picks a yn−1 ∈ F, diﬀerent from Lea’s (last wn−1 ) choices. . . . (The game goes on in this same fashion.) n Lea chooses w1 elements of F. Stan ﬁnally picks a y1 ∈ F, diﬀerent from Lea’s (last w1 ) choices. The winner is Stan if he could pick y = (y1 , . . . , yn ) such that y ∈ V , otherwise Lea wins the game. (Also, if in any step there is no proper choice yi for Stan, then Lea wins.) Example 1. Let n = 5, and α, β ∈ F be diﬀerent elements. Let V be the set of all α-β sequences in F5 in which the number of the α coordinates is 1, 2 or 3. We claim that Lea can win with the question vector w = (11100), but with w = (01110) Stan has a chance to win. Indeed, let w = (11100). To have y ∈ V , Stan is forced to select values from {α, β}. If Stan gives only β for the last 2 coordinates, then Lea will choose α in the ﬁrst three, therefore y cannot contain any α coordinates. However if Stan gives at least one α for the last 2 coordinates, then Lea, by keeping on choosing β, can prevent y to have at least two β coordinates. In the case w = (01110) Stan’s winning strategy is to pick y5 = β, and choose from {α, β} (for the 4th, 3rd and 2nd coordinates). One can easily check that y1 then can always be taken such that y ∈ V .

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It is quite clear that, being a ﬁnite deterministic game, in Lex(V ; w) either Lea or Stan has a winning strategy. We will simply say that Lea or Stan wins Lex(V ; w) accordingly. The main theorem of this section is the following. Theorem 1. Let V ⊆ Fn be a ﬁnite set and w ∈ Nn . Stan wins Lex(V ; w) if and only if xw ∈ Sm(I(V )). An immediate consequence is that Lea wins the game iﬀ xw is a leading monomial for I(V ). There is a fast algorithm3 which lists those w ∈ Nn , for which Stan wins Lex(V ; w) for a given V . In view of Theorem 1, it actually computes the lex standard monomials of I(V ). In this paper we intend to use the Theorem to obtain explicit combinatorial description of Sm(I(V )) for some interesting sets V . Also, note that the game does not use anything more from the properties of the base ﬁeld than its cardinality. That is, we can conclude that the set of lex standard monomials of a vanishing ideal is rather a combinatorial object, than an algebraic one. In line with the recursive nature of the game, we will use induction on n to prove the theorem. The following notation will be useful. For y = (y1 , . . . , yn ) ∈ Fn we set y = (y1 , . . . , yn−1 ), if n ≥ 2. We shall also use y for denoting a vector of length n − 1, even if it is not a preﬁx of a vector of length n. Similarly we shall write sometimes w, or even xw instead wn−1 1 of xw 1 . . . xn−1 . Let y ∈ F, suppose that n ≥ 2, and set

Vy = y ∈ Fn−1 : (y, y) ∈ V . It is clear that if Stan picks yn = y in the ﬁrst step, then they continue as if they have just started a Lex(Vy ; w) game. Proof of Theorem 1. We prove the statement by induction on n. The case n = 1 is easy. Let w ≥ 0 be an integer. Then xw ∈ Sm(I(V )) if and only if w < |Sm(I(V ))| = |V | by the fact that Sm(I(V )) is a downset with respect to division. But this means precisely that there has to be a y ∈ V which is not among Lea’s guesses, thus Stan wins the game by picking that y. Suppose that n ≥ 2, and that the theorem is true for n − 1. Set

Z = y ∈ F : xw ∈ Sm(I(Vy )) . The inductive hypothesis yields that Stan wins Lex(I(Vy ); w) if and only if y ∈ Z. From what we said about the connection between the games Lex(V ; w) and Lex(Vy ; w) it follows that Stan wins Lex(V ; w) if and only if wn < |Z|. Therefore it is enough to show that It uses constant times |V |nk comparisons of ﬁeld elements in the worst case, where k is the maximum number of diﬀerent elements which appear in a ﬁxed coordinate of points of V ; see [12].

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Some Meeting Points of Gr¨ obner Bases and Combinatorics

xw ∈ Sm (I(V ))

⇐⇒

213

wn < y ∈ F : xw ∈ Sm (I(Vy )) .

Suppose ﬁrst that xw ∈ Lm(I(V )), and let f (x) ∈ I(V ) be a witness of this fact, that is lm(f ) = xw . By collecting together the terms of the form xw xin (i ∈ N) we get a decomposition f (x) = xw g(xn ) + h(x), where all monomials of h(x) are lexicographically smaller than xw , and deg(g) = wn . If y ∈ F is not a root of g(xn ), then fˆ(x) = xw g(y)+h(x, y) is a polynomial which vanishes on Vy , and has the property that lm(fˆ) = xw . Thus, if y is not a root of g, then xw ∈ Lm(I(Vy )). In other words there are at most deg(g) = wn elements y ∈ F such that xw ∈ Sm(I(Vy )). For the other direction, assume that xw ∈ Sm(I(V )). First note that by the ﬁniteness of V , we have Vy = ∅ (and then Sm(I(Vy )) = ∅) with ﬁnitely many exceptions y ∈ F, hence |Z| < ∞. Now, it suﬃces to show that |Z| |Z| xw xn ∈ Lm(I(V )), since in this case xw xn cannot be a divisor of xw , that is wn < |Z|. Set F = {y ∈ F : Vy = ∅} and y ∈ F \ Z. On one hand, xw ∈ Lm(I(Vy )) implies the existence of a polynomial fy (x) such that all monomials of f (x) are less than xw , and xw + fy (x) ∈ I(Vy ). On the other hand, let χy (xn ) be a polynomial such that for y ∈ F \ Z 1, y = y, (3) χy (y ) = 0, otherwise. Since F is ﬁnite, such a polynomial does exist. And ﬁnally let w χy (xn )fy (x) · (xn − y). s(x) = x + y∈F \Z

By the properties of the lex order lm(xw +

y∈Z

y∈F

χy (xn )fy (x)) = xw , |Z|

therefore we have that the leading monomial of s(x) is xw xn . It remains to verify s(x) ∈ I(V ). Let y = (y, y) ∈ V be arbitrary. Clearly Vy = ∅, that is y ∈ F . We may suppose that y ∈ Z for otherwise the second term of s(x) vanishes on y. Property (3) of the polynomials χy (xn ) gives (for some α ∈ F) s(x, y) = xw + χy (y)fy (x) · α = xw + fy (x) · α, y ∈F \Z

which vanishes on y ∈ Vy by the deﬁnition of fy , thus s(x) is zero on y. This completes the proof. For those, who do not like playing whilst doing math, we emphasize below the main point of Theorem 1, a fact ﬁrst noted by Cerlienco and Mureddu [8].

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Corollary 1. If V ⊆ Fn is ﬁnite, n ≥ 2, and w ∈ Nn then

xw ∈ Smlex (I(V )) ⇐⇒ wn < y ∈ F : xw ∈ Smlex (I(Vy )) . Theorem 1 has the immediate consequence that the standard monomials are largely independent of the base ﬁeld F and of the precise embedding of V into Fn . As here we consider more than one ﬁeld, let us temporarily put IF (V ) for the polynomial ideal I(V ) in F[x]. Corollary 2. Assume that V ⊆ V1 × · · · × Vn for some ﬁnite sets Vi ⊆ F. Let ˆ are injective maps for i ∈ [n]. Let ˆ be any ﬁeld and suppose that ϕi : Vi → F F ˆ V be the image of V , that is Vˆ = {(ϕ1 (y1 ), . . . , ϕn (yn )) : y ∈ V }. Then Sm(IF (V )) = Sm(IFˆ (Vˆ )). In particular, if V ⊆ {0, 1}n then the set Sm(IF (V )) is independent of the base ﬁeld F. Proof. The Lex(V ; w) game is essentially the same as the Lex(Vˆ ; w) game since we have changed only the names of the elements (bijectively). The second part follows from the ﬁrst, because 0 = 1 in F for any ﬁeld F. The second part of the corollary concerning sets V ⊆ {0, 1}n has been proven in [2] by a diﬀerent method. We now show that the reduced lexicographic Gr¨ obner basis of IF (V ) for a set V ⊆ {0, 1}n is essentially the same over any ﬁeld. We remark that this can be generalized to ﬁnite sets with more than two integer coordinate values. If f ∈ Z[x], then for all ﬁelds F of characteristic 0 we clearly have f ∈ F[x], but also if the characteristic of F is p > 0, we can still consider f as an element of F[x] by reducing its integer coeﬃcients modulo p. obner basis G of IQ (V ) Corollary 3. If V ⊆ {0, 1}n , then the reduced lex Gr¨ has integer coeﬃcients. For an arbitrary ﬁeld F, the set in F[x] corresponding to G is the reduced lex Gr¨ obner basis of the ideal IF (V ). Proof. Let xw +g(x) be an element of the reduced lex Gr¨ obner basis of IQ (V ), where every monomial of g ∈ Q[x] is smaller than xw , and is contained in Sm(IQ (V )). Suppose by contradiction that g ∈ Z[x]. Let z ∈ Z such that zg(x) has relatively prime integer coeﬃcients. If a prime p divides z, then reduce zg ∈ Z[x] modulo p to get a polynomial over Fp . It is a nonzero polynomial which (modulo p) vanishes on V , as zxw + zg(x) vanishes on V and p | z. Thus the leading monomial of zg(x) is in Lm(IFp (V )) = Lm(IQ (V )), by Corollary 2. That is a contradiction. For the second statement, let F be an arbitrary ﬁeld and let us think of G as a subset of F[x]. Obviously G ⊆ IF (V ) is still true and the leading monomials of G remain the same. By Lm(IF (V )) = Lm(IQ (V )), we have that G is a Gr¨ obner basis of IF (V ). As the elements of G, except for their leading monomials, are linear combinations of standard monomials, G is also reduced.

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Before going on to present mathematical (mostly combinatorial) applications of the Lex Game, we brieﬂy comment on the algorithmic problem of actually computing standard monomials, or more generally a basis of Sm(IF (V )) over F. The problem has had a long history starting with the outstanding paper by Buchberger and M¨oller [7]. Their algorithm, as well as the subsequent methods of Marinari, M¨ oller and Mora [21] and Abbott, Bigatti, Kreuzer and Robbiano [1] give also a Gr¨ obner basis of IF (V ). For the arithmetic complexity of these methods we have the bound O(n2 m3 ) when V is a subset of Fn and |V | = m (see Sect. 3 in [10] for a related discussion). The Lex Game provides only the standard monomials, but in return it appears to lead to a much faster algorithm (see [12] for the details). In general we have the bound O(nm2 ). In some important special cases, such as the case of small ﬁnite ground ﬁelds which appear naturally in coding applications, one can even have a linear bound O(nm) on the time demand of the algorithm.

4 Applications 4.1 Generalization of the Fundamental Theorem of Symmetric Polynomials Following [19], we present an easy proof of a theorem by Garsia [17], which is a generalization of the fundamental theorem of symmetric polynomials. The ith elementary symmetric polynomial is xw , σi (x) = w∈{0,1}n deg(xw )=i

provided that 0 ≤ i ≤ n. Later we will also use the complete symmetric polynomial of degree i ≥ 0, which is xw . hi (x) = w∈Nn deg(xw )=i

The fundamental theorem of symmetric polynomials claims that if f (x) is a symmetric polynomial, then it can be written uniquely as a ﬁnite sum u f (x) = αu σ(x) , u∈Nn

n u where αu ∈ F, and σ(x) stands for i=1 σi (x)ui . We intend to prove the following generalization, which was obtained by A. Garsia [17].

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Theorem 2. Any polynomial f (x) ∈ F[x] can be written uniquely as a ﬁnite sum u αw,u xw σ(x) , f (x) = w∈Nn u∈Nn w≤v

where v = (0, 1, . . . , n−1), w ≤ v is understood coordinatewise, and αw,u ∈ F. We need some preparations before the proof. Let z1 , . . . , zn be diﬀerent elements of a ﬁeld and set V = {(zπ(1) , . . . , zπ(n) ) : π ∈ Sn } the set of all permutations of the sequence z1 , . . . , zn . We ﬁrst show that the lexicographic standard monomials of I(V ) are ex. In other words, the minimal lex leading actly the divisors of x2 x23 . . . xn−1 n monomials are of the form xii for i ∈ [n]. Proposition 1. For the set of points V deﬁned above, we have that xw is a lexicographic standard monomial of I(V ) if and only if w ≤ (0, 1, . . . , n − 1). Proof. One can get the lexicographic standard monomials of V using the Lex Game (Theorem 1). Suppose that w ≤ (0, 1, . . . , n − 1). Then Stan’s strategy will be to pick in the (n − i + 1)th step (for yi ) any element from the set {z1 , . . . , zn } \ {yn , . . . , yi+1 }. This set has exactly i elements, so wi < i guarantees that Lea cannot choose all of them, that is there will always be a proper choice for Stan. On the other hand, if for example wi ≥ i, then in the (n − i + 1)th step Lea can choose all the elements of {z1 , . . . , zn } \ {yn , . . . , yi+1 }, thus yi will either be the same as a previously selected yj (and then y ∈ V ) or an element diﬀerent from all zj (again y ∈ V ). We use the following easy fact without proof (see for example [9]) which holds for all i ∈ [n]: i

(−1)j hi−j (xi , . . . , xn )σj (x) = 0.

(4)

j=0

Let i ∈ [n] and set fi (x) =

i (−1)j hi−j (xi , . . . , xn )σj (z). j=0

Proposition 2. The set of polynomials {fi : i ∈ [n]} is the reduced Gr¨ obner basis of V for all term orders, such that the order of the variables is x1 x2 · · · xn .

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Proof. Clearly, if x1 x2 · · · xn holds for a term order, then lm(fi ) = xii . It is also obvious by Proposition 1 that every monomial of fi (x) − xii is a lex standard monomial. Equation 4 implies that fi vanishes on V . As the minimal lex leading monomials (again by Proposition 1) are {xii : i ∈ [n]}, we have obner basis. But the leading that {fi : i ∈ [n]} is indeed a reduced lex Gr¨ monomials of the fi for all term orders ≺ considered in the statement are the same, thus Smlex (I(V )) ⊇ Sm≺ (I(V )). Due to the equality of the cardinalities of the two sides, we have that the standard monomials are the same for all obner term orders considered. We conclude that {fi : i ∈ [n]} is a reduced Gr¨ basis also with respect to ≺. Proof of Theorem 2. We had a good reason for not choosing base ﬁeld for V until now. Let F(z) be the function ﬁeld over F in n variables z1 , . . . , zn and let V ⊆ F(z) be the set of all permutations of these variables, as before. Let f (x) ∈ F[x] ⊆ F(z)[x] be any polynomial, and reduce f (x) by the Gr¨ obner basis {fi (x) ∈ F(z)[x] : i ∈ [n]} of V . The result is an F(z)-linear combination of monomials xw ∈ Sm(I(V )). Furthermore, since actually fi ∈ F[z][x], and fi is symmetric in the variables z1 , . . . , zn , the coeﬃcients of the xw ∈ Sm(I(V )) in this F(z)-linear combination are symmetric polynomials from F[z]. Thus as functions on V , we have an equality f (x) = xw gw (z), xw ∈Sm(I(V ))

where gw (z) ∈ F[z] is a symmetric polynomial. Therefore putting z in the place of x (since z ∈ V ) we get the equation zw gw (z) f (z) = zw ∈Sm(I(V ))

of elements of F(z). An application of the fundamental theorem of symmetric polynomials, together with Sm(I(V )) = {xw : w ≤ (0, 1, . . . , n − 1)} yields the existence of the required form for f . Uniqueness now follows: suppose that u αw,u xw σ(x) . f (x) = xw ∈Sm(I(V )) u∈Nn

Then as functions on V we have u f (x) = αw,u xw σ(z) = xw ∈Sm(I(V

))

u∈Nn

xw ∈Sm(I(V

xw g˜w (z), ))

for some polynomials g˜w (z) ∈ F[z]. We expressed f (x) as an F(z)-linear combination of standard monomials. But this is unique, hence g˜w (z) = gw (z), and so (using the uniqueness part of the fundamental theorem of symmetric polynomials) the claim follows.

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It is instructive to compare our approach here to the one followed by Buchberger and Elias in [6]. They used Gr¨ obner bases to detect and guess identities among polynomials, which involved Fermat polynomials and elementary symmetric polynomials. Subsequently they went on, generalized these to obtain conjectures and then proved these conjectures by traditional inductive means. Here we employ Gr¨ obner bases as a proof technique to establish the generalized identity constituting Theorem 2. 4.2 Wilson’s Rank Formula [n] Consider the inclusion matrix A = I [n] d , m , where m ≤ d ≤ n − m. A famous theorem of Richard M. Wilson [23, Theorem 2] describes a diagonal form of A over Z. A is shown to be row-column equivalent over nZto a d−i with multiplicity ni − i−1 for diagonal matrix with diagonal entries m−i 0 ≤ i ≤ m. As a corollary, the following rank formula holds: Theorem 3. Let p be a prime. Then rankFp (A) =

n n − . i i−1

0≤i≤m d−i p(m−i )

We shall outline a simple proof which uses polynomial functions, and some simple notions related to Gr¨ obner bases. We note ﬁrst that the rank of A is exactly the dimension of the linear space Pd,m over Fp of the functions from V([n]) to Fp which are spanned by the monomials xM with |M | = m. d Let Pm denote the subspace of homogeneous multilinear polynomials in Fp [x] of degree m. Suppose that m ≤ n/2, and for a set M ⊆ [n], |M | ≤ m we deﬁne the multilinear polynomial xM ∈ Pm . yM = M ⊇M |M |=m

To simplify our notation, we write I for the vanishing ideal I

[n] m

of

[n] m .

Lemma 1. The collection of polynomials yM , where xM ∈ Sm(I), is a linear basis of Pm over Fp . Proof. Since {xM + I : xM ∈ Sm(I)} is a linear basis of Fp [x]/I, and xM + I = yM + I (they represent the same function on V([n]) ), we obtain that m

{yM + I : xM ∈ Sm(I)} is a basis of Fp [x]/I. As yM ∈ Pm , it is also clear that Pm + I = Fp [x]. From the fact that Pm ∩ I = {0}, we have a natural isomorphism Pm → Fp [x]/I which sends yM to yM + I. We conclude that {yM : xM ∈ Sm(I)} is indeed a basis of Pm . We can state Wilson’s rank formula in this setting as follows.

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Theorem 4. Let p be a prime, suppose that m ≤ d ≤ n − m and put I = . A basis of the space Pd,m of Fp -valued functions on V([n]) , which are I [n] m Fp -linear combinations of monomials xM , |M | = m is d − |M | . B = yM : xM ∈ Sm (I) , p m − |M |

d

In particular, dimFp Pd,m

n n = |B| = − . i i−1 0≤i≤m d−i p(m−i )

Proof. Let vF be the characteristic vector of a d-subset of [n]. It is immediate that d − |M | · xM (vF ). (5) yM (vF ) = m − |M | We obtain that, as a function on V([n]) , yM is a scalar multiple of xM . This, tod gether with the linear independence of the xM gives that B is an independent set. Also, B spans Pd,m , because Pm spans Pd,m by deﬁnition, and the yM span Pm by Lemma it remains to verify that for 0 ≤ i ≤ m nconclude, 1. To monomials of degree i in Sm(I). This will be there are exactly ni − i−1 proven in Lemma 2. Lemma 2. For an arbitrary term order and any integers 0 ≤ i ≤ m ≤ n monomials of degree i in Sm I [n] . there are exactly ni − i−1 m

n 2,

Proof. We will restrict ourselves to the lex order. Note that this is enough for completing the proof of Theorem 4. The full proof could be carried out by the same ideas we use in Proposition 2 or outline after Theorem 5. We say that a vector w ∈ {0, 1}n is a ballot sequence if in every preﬁx of w there are at least as many 0, as 1 coordinates. We shall prove that xw is [n] a lex standard monomial for I = I m iﬀ deg(xw ) ≤ m and w is a ballot sequence. By Theorem 1, we can use the Lex Game Lex(V([n]) ; w) to show m this. If the number of 1 coordinates in w is more than m, then Lea will choose 0 at each of her guesses. This way, Stan has to put yi = 1 for more than m times, therefore y ∈ V([n]) at the end, and Lea wins. That is, if deg(xw ) > m, m

then xw ∈ Lm(I). Suppose now, that deg(xw ) ≤ m and w is not a ballot sequence. Let i ∈ [n] be such that (w1 , . . . , wi ) is the shortest preﬁx of w that violates the ballot condition. It is easy to see that i is odd, and there are exactly i+1 2 coordinates equal to 1. Assume that when in the game Stan picked yi+1 then there are m − k ones among yn , . . . , yi+1 . Stan would win only if he could pick the remaining yi , . . . , y1 , such that k of them was 1, i − k of them was 0. But if

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i+1 k ≤ i−1 2 , then Lea always chooses 0, thus there will be at least 2 > k ones i−1 i−1 among yi , . . . , y1 . And when k > 2 , then i − k ≤ 2 , so if Lea keeps on choosing 1, then Stan has to claim at least i+1 2 > i − k zero coordinates, and hence he loses the game. Next we show how Stan can win if w is a ballot sequence with at most m ones. Set J = {j ∈ [n] : wj = 1}. For all j ∈ J let us pick an (j) ∈ [n], such that w(j) = 0, (j) < j, and : J → [n] is injective. (This can be done if w is a ballot sequence.) Let us put L = {(j) : j ∈ J}, and K = [n] \ (J ∪ L). Stan’s strategy to choose yi is the following. If i ∈ J, then Lea will guess something, so he just claims the opposite (in {0, 1}). If i ∈ L, say i = (j), then he picks y(j) , such that {yj , y(j) } = {0, 1}. (Note that when choosing the (j)th coordinate, he already ﬁxed yj by (j) < j.) This way, Stan will have exactly |J| ones in (yi : i ∈ J ∪ L). Therefore he picks m − |J| ones from the yk (k ∈ K), and wins. Now it follows immediately, that the lex standard monomials of I [n] m [n] of degree at most i are the same as the lex standard monomials of I . i n standard In particular, there are ni of them, and then there are ni − i−1 monomials of degree i. This proves the lemma.

The approach given here allows a considerable generalization of the rank formula. We present without proof a result of this type (for details, see [16]). · ··
d−i , m2 −i , . . . , md−i . where ni = gcd md−i 1 −i r −i

4.3 Applications to Modulo q -wide Families In this subsection we give two applications of the Gr¨ obner methods to extremal set theory. We prove upper bounds on the cardinality of a family of subsets of [n] with certain restrictions: we will consider modulo q L-intersecting, Lavoiding families, and families that do not shatter large sets. We shall omit a part of the proof, but give the ideas. A detailed proof can be found in [11]. Let us consider the following family of sets. Let q, d, and be integers, such that 1 ≤ < q. Then the modulo q complete -wide family is G = {G ⊆ [n] : ∃g ∈ Z such that d ≤ g < d + , and |G| ≡ g (mod q)} . In other words, G contains all subsets of [n] which have cardinality modulo q in the interval [d, d + − 1] (of length ). The restrictions on the parameters and q tell us exactly that if |G| ≡ d + (mod q), then G ∈ G (that is, G is in fact -wide). Subfamilies of G are called modulo q -wide families. The following theorem will be crucial in both applications.

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Theorem 5. Let p be a prime, and q be a power of p. Denote by HG (m) the Hilbert-function over Fp of a modulo q complete -wide family G. If 0 ≤ m ≤ n+ 2 , then ∞ −1 n HG (m) ≤ . m − jq − k j=0 k=0

A sketch of the proof is the following. One can obtain the lex standard monomials of I(G) by the Lex Game method. Then also the lexicographic Gr¨ obner basis can be constructed: for each minimal lex leading monomial xw , we can exhibit a polynomial fw in the ideal, such that lm(fw ) = xw . It turns out that the lex and deglex leading monomials of these polynomials are the same. From this fact it follows that what we got is a deglex Gr¨ obner basis as well, and that the lex and deglex standard monomials are the same. (This is the same way to compute the deglex Gr¨ obner basis as in Proposition 2.) This is good news, since by counting the deglex standard monomials of degree at most m, we obtain the exact value of HG (m). The formula in Theorem 5 is then a convenient upper bound of that value. One may compare this result to Lemma 2, noting that if q > n and = 1, then the modulo q complete -wide family is just [n] d . 4.3.1 Modulo q L-intersecting, L-avoiding Families Let L be a subset of integers and F be a system of sets. We say that F is modulo q L-avoiding if F ∈ F and f ∈ L implies |F | ≡ f (mod q). We call F modulo q L-intersecting if for any two distinct sets F1 , F2 ∈ F a congruence |F1 ∩ F2 | ≡ f (mod q) holds for some f ∈ L. The maximum number of sets a modulo q L-avoiding, L-intersecting set family can contain has been studied extensively, see [11] for more details. We have the following result in this direction. We call a set L ⊆ {0, . . . , q − 1} a modulo q interval if it is either an interval of integers, or a union of two intervals L1 and L2 , such that 0 ∈ L1 and q − 1 ∈ L2 . Theorem 6. Let q be a power of a prime, L be a modulo q interval and F ⊆ 2[n] be a modulo q L-avoiding, L-intersecting family of sets. If |L| ≤ n − q + 2, then q−1 n |F | ≤ . k k=|L|

The following lemma is left as an exercise for the reader. Lemma 3. If f is an integer, q is a power of a prime p, then 0 (mod p), if f ≡ 0 (mod q) f −1 ≡ q−1 1 (mod p), if f ≡ 0 (mod q).

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Proof of Theorem 6. Put = q − |L|. If L is an interval of integers, then set d = max L + 1, otherwise, when L is the union of two (separate) intervals L1 , L2 and 0 ∈ L1 , set d = max L1 + 1. Denote by G the modulo q complete -wide family with this parameter d. Then by the deﬁnitions F ⊆ G. For any F ∈ F we deﬁne the polynomial fˆF (x) ∈ Q[x] to be ⎛ ⎞ q−1 ⎜ x · vF − k − 1 ⎟ ⎟ reduced by x2i − xi (i ∈ [n]), fˆF (x) = ⎜ ⎝ ⎠ q−1 k=0 k ∈L

n where x · v = i=1 xi vi is the usual scalar product of row vectors. We claim that fˆF ∈ Z[x]. Since we have reduced with x2i −xi , we have that fˆF (x) is multilinear, thus fˆF = G⊆[n] αG xG with some coeﬃcients αG ∈ Q. If fˆF ∈ Z[x], then let G be a minimal set with respect to inclusion, such that αG ∈ Z. Clearly, the reduction with the polynomials x2i − xi does not change the value of the original polynomial on 0-1 vectors, therefore fF (vG ) is an integer. Thus substituting vG we get that fF (vG ) = G G αG + αG , a contradiction since the coeﬃcients αG are integers. We have proven that fˆF ∈ Z[x]. Suppose that q is a power of a prime p and let F ∈ F be a set. Then q−1 |F ∩ F | − k − 1 ˆ fF (vF ) = . (6) q−1 k=0 k ∈L

If F = F , then, as F is modulo q L-intersecting, |F ∩ F | − k cannot be congruent to 0 modulo q for k ∈ L. That is (by Lemma 3), if F = F , then all terms of the sum in (6) are zero modulo p. If F = F , then using that F is modulo q L-avoiding, we have exactly one nonzero term modulo p, which is actually congruent to 1. Write fF for the polynomial in Fp [x] we obtain from fˆF by reducing its integer coeﬃcients modulo p. The above argument yields 0 if F = F , fF (vF ) = 1 if F = F . Since the degree of fˆF is at most q − 1, the same is true for fF as well. Using our earlier notation, this means that fF ∈ Fp [x]≤q−1 . We claim that the images f F of the fF in the quotient space Fp [x]≤q−1 I(G)≤q−1 are linearly independent over Fp . Indeed, suppose that αF f F = 0 (7) F ∈F

for some αF ∈ Fp . The elements of Fp [x]/I(G) are functions on the characteristic vectors of G. In particular (7) still holds if we substitute vF for some F ∈ F ⊆ G. The substitution gives αF = 0 immediately.

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To conclude, note that the number of the polynomials fF is bounded by the dimension of Fp [x]≤q−1 I(G)≤q−1 , that is |F | ≤ dimFp Fp [x]≤q−1 I(G)≤q−1 = HG (q − 1) q−1 ∞ −1 n n ≤ = q − 1 − jq − k k j=0 k=0

k=|L|

by Theorem 5 (which we are allowed to use as |L| ≤ n − q + 2 implies the assumption q − 1 ≤ n+ 2 of the theorem). 4.3.2 Set Families which do not Shatter Large Sets Consider a family F of subsets of [n]. We say that F shatters M ⊆ [n] if {F ∩ M : F ∈ F } = 2M . The system of sets F is an -antichain if it does not contain + 1 distinct sets F0 , F1 , . . . , F such that F0 F1 · · · F . Frankl [14] conjectured that if an -antichain F shatters no set of size m+1 −1 n for some integer 0 ≤ m ≤ n+ k=0 m−k must hold. 2 − 1, then |F | ≤ An -wide family (which of course can be understood as a modulo q -wide family for some q large enough) is an -antichain. In their article [15], Friedl, Heged˝ us and R´onyai showed that the upper bound is valid for -wide families. The next theorem is a generalization of that result, the special case follows by choosing q > n. Theorem 7. Let F ⊆ 2[n] be a modulo q -wide family of sets, where q is a prime power. If F shatters no set of size m+1 for some integer 0 ≤ m ≤ n+ 2 , then ∞ −1 n |F | ≤ . m − jq − k j=0 k=0

Proof. We ﬁrst prove that if xM is a standard monomial of any set system F , then F shatters M . Suppose that N ⊆ M , but N ∈ {F ∩ M : F ∈ F }. Let v = vN be the characteristic vector of N . Then the polynomial (xi + vi − 1) i∈M

vanishes on VF and its leading monomial is xM , thus xM ∈ Lm(I(F )). We conclude that xM ∈ Sm(I(F )) implies |M | ≤ m for a family F which does not shatter any set of size m + 1. Recall that F ⊆ G, where G is a modulo q complete -wide family. This gives Sm(I(F )) ⊆ Sm(I(G)), and so we can bound the cardinality of the standard monomials of F with the number of standard monomials of G of degree at most m. This latter is exactly HG (m), if we consider a degree compatible

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term ordering. (Actually, in this case, we can take any term order, see the discussion after Theorem 5.) Therefore |F | = |Sm (I(F ))| ≤ HG (m), and hence Theorem 5 gives the desired bound.

The inequality in Theorem 7 is sharp. Choose d = m − + 1 for a modulo [n] . Then the fact that F q complete -wide family G, and put F = G ∩ ≤m does not contain any set of size m + 1 implies that it cannot shatter any set ∞ −1 n . of cardinality m + 1. The size of F is precisely j=0 k=0 m−jq−k 4.4 Harima’s Theorem for Set Families Here we prove an important special case of a theorem by T. Harima. It establishes a connection among the Hilbert functions of complementary set families. Theorem 8. Suppose F ⊆ 2[n] and G = 2[n] \ F are nonempty set families. Then for their Hilbert functions we have m n = |G| + HF (m) − HG (n − 1 − m) i i=0 for every m = 0, 1, . . . , n. Theorem 8 was proved by Tadahito Harima for much more general point sets. In formula (3.1.5) of [18] the result is given for two disjoint ﬁnite point sets X, Y ⊂ Pn (F) in the projective n-space over F, instead of VF and VG , such that X ∪ Y is a complete intersection. The formula was used in his characterization of the Hilbert functions of Artinian Gorenstein algebras with the weak Stanley property. Here we focus on 0,1-vectors only. Our approach is based on direct computations with polynomial functions. Proof. For a subset M ⊆ [n], let M c stand for the set [n] \ M . We claim that a monomial xM is a leading monomial for I(F ) if and only if xM c is a standard monomial for I(G). Among the monomials of the form xM , the number of leading monomials for I(F ) is the same as the number of standard monomials for I(G), namely 2n − |F | = |G|, hence the claim will follow if we show that xM ∈ Lm(I(F )) implies xM c ∈ Sm(I(G)). Indeed, suppose for contradiction that we have polynomials f ∈ I(F ) and g ∈ I(G) with leading terms xM and xM c , respectively. Then f · g vanishes on V2[n] and its leading term is x[n] . This is impossible, because |Sm(I(2[n] ))| = 2n = |{xM : M ⊆ [n]}| implies that every multilinear monomial is a standard monomial for V2[n] . Let ≺ be a degree compatible term order on F[x]. Now the number of multilinear leading monomials of degree i for I(F ) is ni −(HF (i)−HF (i−1)).

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By the claim above, this is HG (n − i) − HG (n − i − 1), the number of standard monomials of degree n − i for I(G). We have n = HF (i) − HF (i − 1) + HG (n − i) − HG (n − i − 1), i for every 0 ≤ i ≤ n (we use the convention HF (−1) = HG (−1) = 0). By adding these up for i = 0, . . . , m, we obtain m n i=0

i

= HF (m) + HG (n) − HG (n − m − 1).

The theorem follows now from HG (n) = |G|.

Theorem 8 allows us to formulate an interesting min-max relation. Let F ⊂ 2[n] be a family diﬀerent from ∅ and 2[n] . Let a(F ) stand for the smallest degree of a nonzero multilinear polynomial from F[x] which vanishes on VF . We have 1 ≤ a(F ) ≤ n. Also, we deﬁne b(F ) to be the smallest integer k such that HF (k) = |F |. In other words, b(F ) is the smallest degree k such that every function from VF to F can be represented by a polynomial from F[x] of degree at most k. We have 0 ≤ b(F ) ≤ n. It is easily seen that any polynomial χv ∈ F[x] which is 1 on the vector v ∈ {0, 1}n , and 0 on all other vectors from {0, 1}n must have degree at least n. From that we readily infer that a(F ) + b(2[n] \ F ) ≥ n.

(8)

Theorem 8 implies that, in fact, we have an equality here. Corollary 4. Let F ⊂ 2[n] and G = 2[n] \ F . Assume that both F and G are nonempty. Then we have a(F ) + b(G) = n. Proof. We apply Theorem 8 with m = a(F ) − 1. Note ﬁrst, that m ≥ 0 and HF (m) = H2[n] (m), because the multilinear monomials of degree ≤ m are linearly independent over F, as functions on VF . Theorem 8 gives now that HG (n − m − 1) = |G|, hence b(G) ≤ n − m − 1 = n − a(F ). This, together with (8) proves the assertion. In [22] Theorem 8 is proved over more general coeﬃcient rings, rather than ﬁelds, which include the rings Zk = Z/kZ, where k is a positive integer. An application to the (modular weak degree) complexity of Boolean functions is also given there.

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Acknowledgements Part of this work was done during the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz, Austria. The authors are pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester for their hospitality and attention to this project. Research supported in part by OTKA grants NK72845, NK63066 and K77476.

References 1. J. Abbott, A. Bigatti, M. Kreuzer, and L. Robbiano, Computing ideals of points, J. Symbol. Comput., 30 (2000), 341–356. 2. R. P. Anstee, L. R´onyai, and A. Sali, Shattering news, Graphs Comb., 18 (2002), 59–73. 3. B. Buchberger, Ein Algorithmus zum Auﬃnden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Doctoral thesis, University of Innsbruck, 1965. English translation: An algorithm for ﬁnding the basis elements in the residue class ring modulo a zero dimensional polynomial ideal. J. Symbol. Comput., Special Issue on Logic, Mathematics, and Computer Science: Interactions, 41 (2006), 475–511. 4. B. Buchberger, Ein algorithmisches Kriterium fur die L¨ osbarkeit eines algebraischen Gleichungssystems, Aequationes mathematicae, 4 (1970), 374–383. English translation: An algorithmic criterion for the solvability of algebraic systems of equations, in B. Buchberger, F. Winkler (eds.) Gr¨ obner Bases and Applications, London Mathematical Society Lecture Note Series, Vol. 251, pp. 535–545, Cambridge University Press, Cambridge, 1998. 5. B. Buchberger, Gr¨ obner-Bases: An algorithmic method in polynomial ideal theory, in N. K. Bose (ed.) Multidimensional Systems Theory – Progress, Directions and Open Problems in Multidimensional Systems Theory, pp. 184–232, Reidel, Dordrecht, 1985. Chapter 6. 6. B. Buchberger and J. Elias, Using Gr¨ obner bases for detecting polynomial identities: a case study on Fermat’s ideal, J. Number Theory, 41 (1992), 272–279. 7. H. M. M¨ oller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, in Computer Algebra (1982), Lecture Notes in Comput. Sci., Vol. 144, pp. 24–31, Springer, Berlin, 1982. 8. L. Cerlienco and M. Mureddu, From algebraic sets to monomial linear bases by means of combinatorial algorithms, Discrete Math., 139 (1995), 73–87. 9. D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer, Berlin, 1992. 10. J. B. Farr and S. Gao, Computing Gr¨ obner bases for vanishing ideals of ﬁnite sets of points, in Applied Algebra, Algebraic Algorithms and Error-

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correcting Codes, Lecture Notes in Comput. Sci., Vol. 3857, pp. 118–127, Springer, Berlin, 2006. 11. B. Felszeghy, G. Heged˝ us, and L. R´onyai, Algebraic properties of modulo q complete -wide families. Combinatorics, Probability and Computing, 18 (2009), 309–333. 12. B. Felszeghy, B. R´ath, and L. R´onyai, The lex game and some applications, J. Symbol. Comput., 41 (2006), 663–681. 13. B. Felszeghy, and L. R´ onyai, On the lexicographic standard monomials of zero dimensional ideals, in Proc. 10th Rhine Workshop on Computer Algebra (RWCA), pp. 95–105 (2006). 14. P. Frankl, Traces of antichains, Graphs and Combinatorics, 5 (1989), 295– 299. 15. K. Friedl, G. Heged˝ us, and L. R´onyai, Gr¨ obner bases for complete -wide families, Publ. Math. Debrecen, 70 (2007), 271–290. 16. K. Friedl and L. R´onyai, Order shattering and Wilson’s theorem, Discrete Math., 270 (2003), 127–136. 17. A. M. Garsia, Pebbles and expansions in the polynomial ring, in Polynomial Identities and Combinatorial Methods, Lecture Notes in Pure and Appl. Math., Vol. 235, pp. 261–285 (2003). 18. T. Harima, Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property, Proc. Amer. Math. Soc., 123 (1995), 3631–3638. 19. G. Heged˝ us, A. Nagy, and L. R´onyai, Gr¨ obner bases for permutations and oriented trees, Ann. Univ. Sci. Budapest, Sectio Computatorica, 23 (2004), 137–148. 20. G. E. Moorhouse, Approaching some problems in ﬁnite geometry through algebraic geometry, in M. Klin et al. (eds.) Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, pp. 285–296, Springer, Berlin, 2009 (this volume). 21. M. G. Marinari, H. M. M¨oller, and T. Mora, Gr¨ obner bases of ideals deﬁned by functionals with an application to ideals of projective points, Appl. Algebra Engrg. Comm. Comput., 4 (1993), 103–145. 22. D. Pint´er and L. R´ onyai, On the Hilbert function of complementary set families, Ann. Univ. Sci. Budapest, Sectio Combinatorica, 29 (2008), 175– 198. 23. R. M. Wilson, A diagonal form for the incidence matrices of t-subsets vs k-subsets, Europ. J. Combin., 11 (1990), 609–615.

A Construction of Isomorphism Classes of Oriented Matroids Ralf Gugisch Mathematisches Institut, University of Bayreuth, 95440 Bayreuth, Germany. [email protected]

Summary. We developed a computer program for generating all oriented matroids corresponding to a prescribed underlying matroid. The generation process and its output can be controlled via a variety of possible restrictions allowing to generate speciﬁc sets of oriented matroids. The tool was mainly intended for application in chemistry: According to an idea of A. Dreiding and A. Dress, oriented matroid generation may be used as a ﬁrst step towards a conformation generation for chemical structures. We describe the main ideas of the generation algorithm as well as an application to the conformation analysis of cyclohexane. A combination with the Gr¨ obner base approach for conformation analysis going back to P. Hazebroek and L. Oosterhoﬀ is suggested.

Key words: Generation, Oriented matroid, Chirotope, Aﬃne point conﬁguration, Order type, Conformation, Cyclohexane

1 Motivation: Conformation Spaces in Chemistry As a motivation we want to consider the chemical compound cyclohexane C6 H12 . It consists of six carbon atoms (C) and 12 hydrogen atoms (H), each carbon atom is attached to two hydrogen atoms and the carbon atoms form a 6-ring, see Fig. 1. As usual in organic chemistry and for sake of simplicity, the hydrogen atoms are not taken into consideration any further. The atoms of chemical compounds often appear in more than one possible geometric arrangement, called conformations. In Fig. 2 you can see three different conformations of cyclohexane, namely the chair form and two twisted forms (which are mirror images of each other). However, the number of reasonable conformations is mostly very big. The conformation space of cyclohexane consists of inﬁnitely many conformations similar to or somewhere “in between” the shown ones.

M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 7,

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Fig. 1. The chemical compound cyclohexane

Fig. 2. Conformations of cyclohexane: The chair form (a) and two twisted forms (b, c)

Fig. 3. The Gr¨ obner base approach for analyzing the conformation space of cyclohexane

The conformation space of cyclohexane was analyzed using techniques which may be regarded as predecessors of Gr¨ obner bases by Hazebroek and Oosterhoﬀ [15]. Since then, this example is often used for demonstrating the use of Gr¨ obner bases, e.g. in [16] or in [33]. Assuming standard uniform bond length 1 and bond angles α = acos(− 13 ) ≈ 109◦ (this is the angle between the middlepoint of a regular tetrahedron and two of its vertices respective the carbon atom and two of its four neighbor atoms), one derives a system of polynomial equations from the cyclic structure of the carbon atoms in the following way: Let a1 , . . . , a6 ∈ R3 be the bond-vectors in a given conformation of cyclohexane, i.e. the vectors between two adjacent carbon atoms respectively, oriented in cyclic way, see Fig. 3. The ﬁxed bond lengths and angles lead to quadratic equations using the norm and the inner product of bond vectors: ai = a2i,1 + a2i,2 + a2i,3 = 1, for i = 1, . . . , 6, and

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1 ai , aj = ai,1 aj,1 + ai,2 aj,2 + ai,3 aj,3 = cos α = − , 3 where i and j refer to consecutive vectors. Further, the cyclic structure is encoded by the following vectorial equation: a1 + · · · + a6 = 0, expanding to three equations in the coordinates. Finally, the following equations ﬁx the position of the structure in space: a1,1 = 0,

a1,2 = 0,

a2,1 = 0.

This algebraic system of linear and quadratic equations can be solved easily using Gr¨ obner bases. A more symmetric approach solves a substituted system of linear equations using the variables Sij := ai , aj and some additional polynomial equations coming from Gram determinants in order to enforce three-dimensional solutions. It turns out that the solutions depend on only three independent variables S13 , S35 and S51 , and that the set of triples (S13 , S35 , S51 ) ∈ R3 leading to feasible conformations decomposes into two connected components: a singular point leading to the chair form, and a closed curve leading to ﬂexible conformations containing both twisted forms. Thus under the assumption that the bond lengths and bond angles are preserved, the chair form and the two twisted forms lie in two diﬀerent connected components of the conformation space. There is an extension [21] of this approach to cycloheptane C7 H14 . However, to the knowledge of the author, larger molecular compounds like hydrocarbonic nine- or ten-rings, or more complex structures containing a double bond were not successfully analyzed with this approach, yet, though posted as interesting problem in [23]. In this paper, we want to demonstrate a complementary approach using combinatorial tools. We will not get as detailed information about the structure of the conformation space as the connected components. Instead, we just try to provide a proper set of sample conformations helping to get an overview over the whole space. For practical chemical purposes as the elucidation of structure-property relationships this is quite convenient: One often considers a chemical compound as a mixture of diﬀerent conformations. Of course, the sample set should be of reasonable size and reasonably distributed over the conformation space. Thus, we are faced with the problem of conformation generation, whereby we need to keep in mind, that we do not generate all conformations rather than a (more or less) proper set of sample conformations. One possibility to get such a sample set is to divide the conformation space into proper equivalence classes and to take a representant out of each of the classes. The idealistically best classiﬁcation would be into “watersheds” of a proper energy function, as mentioned in [6]: Two points belong to the same equivalence class if and only if a steepest descents path on the energy

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function starting in either of the two points ends in the same sink. However, energy functions of molecular structures are very complex having a lot of diﬀerent sinks and small watersheds, such that both calculation of all sinks and calculation of watersheds appear not to be feasible. A much coarser division into equivalence classes very common in practice is the one into conﬁgurational stereoisomers. (Two conﬁgurational stereoisomers are distinguished by diﬀerent conﬁgurations at stereocenters, i.e. at carbon atoms having – sloppily circumscribed – four diﬀerent neighbors.) Generation of all conﬁgurational stereoisomers of a chemical compound may be done by a computer very eﬃciently using Nourse’s algorithm [24–26]. Cyclohexane, for example has no stereocenters, i.e. there is only one conﬁgurational stereoisomer of cyclohexane. It turns out that the distinction of stereoisomers is too coarse for the purpose of conformation analysis. Commercial conformation generators as TRIPOS’ Confort or academic ones like Frog [20] additionally generate conformations which are distinguished by standard torsion angles of butane substructures (i.e. four consecutively connected carbon atoms). For discussion of further similar approaches, see [29]. As long as we consider only three standard classes for the classiﬁed torsion angles, namely around the standard torsion angles +60◦ , −60◦ and 180◦ , we can unify and generalize the two classiﬁcation aspects mentioned above using the ideas of Dreiding and Dress [7, 8], see also [31]: Certain oriented matroids, which are in strong relation to the orientation function of the atoms (as points in space), serve well for classiﬁcation and description of conformations. This approach and the necessary mathematical structures are described in Sect. 2. According to the sketched ideas, we promote the following strategy for conformation generation in two steps: 1. Combinatorial level: Generate all oriented matroids which are feasible (by means of simple combinatorial tests) for a chemical conformation. 2. Geometric level: Try to ﬁnd a feasible conformation as (aﬃne) realization for each of these oriented matroids. (Note that not for each oriented matroid there exist aﬃne realizations. Though the problem of deciding whether a chirotope is aﬃnely realizable and of ﬁnding a realization is known to be NP-hard, there exist algorithms which are suitable for small sizes [30, 28].) In [14] the computer program origen was presented, a generator of oriented matroids serving for step one. In Sect. 3, we describe the features of origen and sketch some algorithmic ideas thereof. Note that there does not yet exist a sophisticated and adjusted solution for the second step of ﬁnding feasible conformations. Nevertheless we are able to demonstrate the conformation generation according to the promoted strategy at hand of the example cyclohexane in Sect. 4.

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2 Oriented Matroids, Chirotopes and Aﬃne Point Conﬁgurations One occurrence of oriented matroids [3, 4] is in connection with studies on sets of points in d-dimensional euclidean space [13, 17]. To any sequence of d + 1 aﬃnely independent points is assigned an orientation (positive or negative). One can determine the orientation of the points p1 , . . . , pd+1 having coordinate vectors pi = (pi,1 , . . . , pi,d ) by calculating the sign of the determinant: ⎞ ⎛ 1 ··· 1 ⎟ ⎜ ⎜ p1,1 · · · pd+1,1 ⎟ ⎟ ⎜ ⎟ ori(p1 , . . . , pd+1 ) = det ⎜ ⎜ .. .. ⎟ . ⎜ . . ⎟ ⎠ ⎝ p1,d · · · pd+1,d For example, in two-dimensional plane, a sequence of three non-collinear points is oriented either anticlockwise (positive) or clockwise (negative). In three-dimensional space, the orientation of four points p1 , p2 , p3 , and p4 in arbitrary position can be determined by the common “right-hand rule”: Identifying the point p1 with the wrist of the right hand, the four points have positive orientation if and only if it is possible to point with thumb, index and middleﬁnger in direction to the points p2 , p3 and p4 without overstretching the middle ﬁnger. By this concept of orientation, we assign to any sequence of n points spanning the whole d-dimensional euclidean space an orientation function χ : nd+1 → {0, ±1}, where the function value 0 means, that the corresponding (d + 1)-tuple of points lies in a hyperplane. Here and below n denotes the nelement set {1, 2, . . . , n}. Obviously, the function χ is alternating. We can write χ as sequence of its function values at the ordered (d + 1)-tuples, using the reverse lexical order for listing the tuples. Example 1. Let us consider the following six points in space (see Fig. 4). p(1) = (4, 3, 2), p(4) = (2, 1, 2),

p(2) = (5, 2, 1), p(5) = (1, 2, 3),

p(3) = (4, 1, 2), p(6) = (2, 3, 2).

12 1234 1235 1345 2345 1245 1236 1346 2346 1246 1356 2356 1456 2456 3456 56

The orientation function as sequence of signs for each quadruple of points is:

χ =++0 −−++0 −++0 +++ For example, the function value χ(1, 2, 3, 4) = +1 means, that the four points p(1) , p(2) , p(3) and p(4) are positively oriented, while χ(1, 2, 4, 5) = 0 denotes, that the points p(1) , p(2) , p(4) and p(5) are coplanar.

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Fig. 4. Six points in space

As determinant functions, the orientation functions fulﬁll the binary Grassmann-Pl¨ ucker relations: Let k := d + 1. For any a, b ∈ nk the following holds: χ(a) · χ(b) = +1 =⇒ ∃i ∈ {1, . . . , k} : χ(bi , a2 , . . . , ak ) · χ(b1 , . . . , a1 , . . . , bk ) = +1. (GP) ↑ ith position

(When calling tuples a with χ(a) = 0 bases, we can interpret the GrassmannPl¨ ucker relations as an oriented version of the base exchange axiom known from linear algebra.) In general, alternating, non-trivial (i.e. not constantly zero) functions χ : nk → {0, ±1} fulﬁlling (GP) are called chirotopes of rank k. Chirotopes imply a huge amount of further structure which is embraced in the concept of oriented matroids. For the purpose of this work, it is enough to know that the oriented matroids are in one-to-one correspondence to the pairs {χ, −χ} of chirotopes. From each oriented matroid, we get an unoriented matroid, the underlying matroid, by taking the domain of the chirotope as bases. The oriented matroid is called uniform, if the chirotope has no zero function values (i.e. if the underlying matroid is uniform). The orientation function of a sequence of n spanning points in d-dimensional euclidean space is a chirotope of rank k = d + 1 and thus realizes an oriented matroid. The corresponding oriented matroid is uniform if and only if the sequence of points is in general position, i.e. if each (d + 1)-subset is aﬃnely independent. (If the points are not spanning, then the orientation function is the zero function.) Note that not each chirotope is an orientation function. We call oriented matroids aﬃnely realizable, if a corresponding chirotope is the orientation function of a sequence of points in euclidean space. We call spanning sequences of points in d-dimensional space combinatorially equivalent, or of same signed conﬁguration, if they have the same orientation function. Further, we call spanning sets of points combinatorially equivalent, if there exist numberings of the sets, such that the correspond-

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Fig. 5. The chair form of cyclohexane, with numbered atoms

ing sequences are combinatorially equivalent. The corresponding equivalence classes of sets of points we call signed aﬃne point conﬁgurations. Let χ be the orientation function of a spanning sequence of points. Then the mirror image of the point sequence has −χ as orientation function (i.e. all function values of χ are negated). We call a point set achiral, if the point set and its mirrored point set belong to the same signed conﬁguration. Otherwise the point set is chiral. The two signed conﬁgurations of a chiral point set and its mirrored point set are called enantiomeric to each other. The equivalence classes of point sets arising by identifying chiral signed point conﬁgurations with their enantiomeric conﬁgurations, respectively, we call unsigned aﬃne point conﬁgurations. (Unfortunately, in literature the term (aﬃne) point conﬁguration is used both for signed as well as for unsigned point conﬁgurations. Thus we decided to explicitly distinguish both deﬁnitions. Unsigned point conﬁgurations are often called the order type of a point set, too. See also the discussion of isomorphism aspects of oriented matroids below.) A lot of geometrical applications like convex hulls or triangulations can be realized by solely considering the purely combinatorial information of aﬃne point conﬁgurations, and so it is no surprise that these structures (or equivalent ones) were studied intensively in literature, especially in 2-dimensional case; see for example [13, 17, 18]. Example 2. If we number the atoms of a chemical structure, then each conformation corresponds to a sequence of points in space and thus to a signed point conﬁguration. For example, if we number the atoms of the chair form of cyclohexane as shown in Fig. 5, then the atoms have the same signed conﬁguration as the points in Fig. 4. Thus the chirotope χ from Example 1 is assigned to the chair form of cyclohexane. 2.1 Isomorphism

12 1234 1235 1345 2345 1245 1236 1346 2346 1246 1356 2356 1456 2456 3456 56

Reorderings of a sequence of points in euclidean space lead to diﬀerent orientation functions. For example, if we exchange the labels at the points p(5) and p(6) in Example 1, we get the chirotope χ := χ ◦ (5, 6):

χ =+++0 −+0 −−−−0 −−− If the order of the sequence of points is not important, i.e. when considering sets of points, we need to identify the corresponding chirotopes for all

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reorderings. Two chirotopes are isomorphic with respect to relabeling, if they can be obtained from each other by a permutation of n: ∼ χ ⇐⇒ ∃π ∈ Sn : ∀a ∈ nk : χ (a1 , . . . , an ) = χ(π(a1 ), . . . , π(an )). χ = It is not trivial to recognize whether two chirotopes are isomorphic with respect to relabeling, or not. In [14], an algorithm generating a canonic form for each isomorphism class of chirotopes is described. This algorithm exploits similar ideas as the iterated reﬁnement algorithm of B. McKay for graphs and digraphs [22]. In general, one may consider chirotopes as generalizations of simple digraphs with no (anti-)parallel edges, as the adjacency matrix of such a digraph is a mapping n2 → {0, ±1}, and the generalization is towards mappings nk → {0, ±1}. There are further relevant isomorphisms for chirotopes: Considering the mirror-image of a set of points leads to the negative chirotope −χ, i.e. each function value inverts its sign. In other geometric applications of chirotopes as polytopes and zonotopes, reorientations are playing a role, too. A reorientation at label i changes the signs for all those tuples containing i. For a more detailed explanation of reorientation symmetry, refer to the standard literature on oriented matroids [3, 4]. Thus when generating chirotopes, three kinds of isomorphy may be considered: • relabeling (i.e. permutations of the labels), • negation (i.e. identifying χ and −χ), • reorientation (has no obvious interpretation for aﬃne point conﬁgurations). Note that generation of oriented matroids is equivalent to generation of chirotopes up to negation. For chemical compounds, any permutation of labels acts on the molecular graph, too. Thus relabellings of the atoms of a chemical structure lead to isomorphic conformations if and only if the corresponding permutation is a graph automorphism of the molecular graph. In other words, when we are interested in isomorphic conformations, we need to consider relabellings of the chirotope under action of the graph automorphism group only. 2.2 Radon Partitions and Oriented Circuits An important structure connected to oriented matroids and thus induced by the chirotope are the (oriented ) circuits: A pair (C + , C − ) of two disjoint subsets of n is called a circuit, if there exists an ordered (k + 1)-tuple (c1 , . . . , ck+1 ) and an ∈ {±1}, such that C + ∪ C − ⊆ {c1 , . . . , ck+1 } and for each i = 1, . . . , k + 1: ⎧ + · (−1)i if ci ∈ C + , ⎪ ⎪ ⎨ χ(c1 , . . . , ci−1 , ci+1 , . . . , ck+1 ) = − · (−1)i if ci ∈ C − , ⎪ ⎪ ⎩ 0 else.

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Fig. 6. A minimal Radon partition {+a, −b, +c, +d, −e}

The circuits of an oriented matroid can be obtained from the chirotope by checking all ordered (k + 1)-tuples. We write circuits as signed sets, i.e. we attach the appropriate sign to the elements of C + resp. C − and write it as a set of signed elements. For example, {+1, −3, +4, −6} is a circuit of the chirotope given in Example 1. In an aﬃne point set, two disjoint subsets of points whose convex hulls intersect are called a Radon partition. The circuits of the assigned oriented matroid denote exactly the minimal Radon partitions. Example 3. Consider the ﬁve points a, b, c, d and e in three-dimensional space shown in Fig. 6. The convex hulls of {a, c, d}, a triangle, and of {b, e}, a line, do intersect. The pair of sets is minimal with this property, i.e. deleting one point from either set would lead to non-intersecting convex hulls. Thus, {+a, −b, +c, +d, −e} is a minimal Radon partition. Note that an aﬃnely realizable oriented matroid cannot have positive circuits, i.e. circuits (C + , C − ) with C − = ∅. In other words, absence of positive circuits is a necessary criterion for aﬃne realizability. Chirotopes having no positive circuits are often called acyclic. In addition, Radon partitions, respective (oriented) circuits, serve as useful language to formulate necessary conditions for chemical feasible conformations. See Sect. 4. 2.3 Partial Chirotopes When considering conformations of molecular structures, the orientations of quadruples of atoms are not all of same importance. For example, the orientation of four non-connected, distant atoms is of minor importance for conformational classiﬁcation, while the orientations of the four neighbor atoms of the stereocenters classify the stereoisomers. In order to take this into consideration, the concept of partial chirotopes is useful. A partial chirotope is a mapping from a subset of nk to {0, ±1} being alternating and fulﬁlling (GP) where applicable. A partial chirotope is extendable, if one can extend it to a chirotope. Note that testing extendability for partial chirotopes is, though NP-complete [32], manageable for small n

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(e.g. using origen; already [6] presented a generation algorithm suitable for this task). One can interpret the extendable partial chirotopes as equivalence classes of chirotopes, namely the sets of extensions.

3 The Generator origen An obvious strategy for generation of aﬃne signed point conﬁgurations over n points is to generate all isomorphism classes of chirotopes up to relabeling over n points and to ﬁnd aﬃne realizations in a second step. For unsigned point conﬁgurations, one can generate chirotopes up to relabeling and negation – or oriented matroids in one of its other representations. (Recall that oriented matroids are in one-to-one correspondence to sets {χ, −χ} of chirotopes and their negatives.) The latter was done by Aichholzer, Aurenhammer and Krasser in the two-dimensional case [1, 2] as well as by Finschi and Fukuda for arbitrary dimension [10, 11]. The generator of uniform oriented matroids by Bokowski and de Oliveira [5] is applicable to this task, too. However, generating full catalogs of point conﬁgurations is limited to small numbers of points. In Table 1, you see the numbers of three-dimensional unsigned point conﬁgurations (including degenerated cases) for n ≤ 8 points, as computed in [10] and approved by origen. It does not make much sense to continue storing full catalogs for larger n’s. We want to exploit restrictions already during the generation process and speciﬁcally compute conﬁgurations e.g. for diﬀerent chemical compounds. Generation of partial chirotopes in the sense of partial two- or threedimensional signed point conﬁgurations was discussed in [34] in a chemical context. Generation up to relabeling with respect to an automorphism group of some additional structure (the molecular graph) was considered, too. We developed the generator origen generating all oriented matroids (in form of chirotopes) corresponding to a prescribed set of parameters and restrictions. The oriented matroids are represented as chirotopes. The generation process and its output can be controlled via a variety of possible parameters and restrictions: • The parameters n and k specify size (number of points) and rank (k = d + 1) of the chirotopes. • Properties as simple or cosimple underlying matroids, acyclic chirotopes (i.e. no positive circuits) can be speciﬁed. • The domain D ⊆ nk of the chirotope (i.e. the underlying matroid) as well as single orientations can be prescribed. Table 1. Unsigned 3D point conﬁgurations for small numbers of points (from [10]) Number of points Point conﬁgurations

4 1

5 5

6 55

7 5083

8 10 775 236

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• There is the possibility to specify a list of forbidden circuits. • You can specify a subset R ⊆ nk of relevant k-tuples. Only extendable partial chirotopes on R are generated. (Actually, origen generates a transversal of extensions.) • Diﬀerent kinds of isomorphisms for chirotopes can be combined arbitrarily: relabeling, negation and reorientation. • For relabeling isomorphy, the acting group can be restricted to a subgroup G of the symmetric group Sn . • A group A of relabeling automorphisms can be prescribed, such that each generated solution has A as subgroup of its automorphism group. The generator is available in internet, see http://www.mathe2.uni-bayreuth.de/ralfg/origen.php 3.1 Notes on the Generation Algorithm An important well-known fact for eﬃcient validation of chirotopes during generation process is, that we do not need to test all n2k Grassmann-Pl¨ ucker relations rather than only the ones belonging to a pair a, b of k-tuples diﬀering in exactly two points (the so-called three-term Grassmann-Pl¨ ucker relations). k → {0, ±1}, it suﬃces Thus having an alternating, non-trivial function χ : n n k+2 · 4 conditions (see [14]): to test the following k+2 For each (k + 2)-subset X of n, and for each 4-subset {a1 , a2 , b1 , b2 } thereof, with a1 < a2 < b1 < b2 , let x be the ordered (k − 2)-sequence of X \ {a1 , a2 , b1 , b2 ). Using the abbreviations1 a := (x, a1 , a2 ) b := (x, b1 , b2 )

a := (x, b1 , a2 ) b := (x, a1 , b2 )

a := (x, b2 , a2 ) b := (x, b1 , a1 )

and s1 := χ(a) · χ(b)

s2 := χ(a ) · χ(b )

s3 := χ(a ) · χ(b ),

we need to test (s1 = 0 ∧ (s2 = −s3 )) ∨ (s1 = 0 ∧ (s1 = s2 ∨ s1 = s3 ))

(GP3 )

The generation is split into two main levels. In the ﬁrst main level, all underlying matroids satisfying the given restrictions are generated (i.e. the possible domains for the chirotopes), and in the second main level, orientations are distributed to a given underlying matroid. This splitting is beneﬁcial according to the homomorphism principle [19]. For a (k − 2)-tuple x ∈ nk−2 and i, j ∈ n we write (a, i, j) as the elongation of x to a k-tuple. 1

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Fig. 7. Scheme of the backtrack generating algorithm

In both main levels, we generate mappings on nk : The underlying matroid is represented as symmetric function nk → {0, 1} specifying the bases, and the chirotope itself is an alternating function nk → {0, ±1} as described above. Thus in both cases it suﬃces to specify the function values at increasing n k-tuples, and we implement both steps as backtrack algorithms of depth functions k , see Fig. 7. Principally, we generate this way all alternating χ : nk → {0, ±1} in a global backtrack algorithm of depth 2 · nk . By pruning branches of this tree due to several kinds of tests on each level, we ensure, that only chirotopes with the requested properties are reached at the last level. In either main generation step, denote the root of the (local) backtrack tree as level −1. Then, on level i, the function value χ(a(i) ) for the ith ordered (i) (i) k-tuple a(i) = (a1 , . . . , ak ) ∈ nk (using the reverse lexical order on k-tuples) is speciﬁed. In case of the matroid generation (ﬁrst main level) there is the choice between 0 (no base) and 1 (base), in the second generation step for given underlying matroid there is a choice for bases only, namely to choose their orientations (−1 or +1). The following rules lead to prunings: • If a value χ(a) is prescribed due to the generation input, then the alternative choice is skipped. • If a group A ≤ Sn of automorphisms is prescribed, then we calculate the orbits on the k-subsets in advance. After the ﬁrst function value of an k-tuple corresponding to an orbit is speciﬁed, the remaining values ensue. • On level nk , n ≤ n, we completed a mapping on nk . Test if the restricted mapping on nk is canonic. The last point guarantees that each chirotope is generated only once up to considered symmetry aspects. The idea behind is the principle of orderly generation [9, 27]. It works only if the deﬁnition of the canonic form is compatible with the generation strategy in the following sense: For a canonic chirotope χ : nk → {0, ±1}, each restriction to nk , n ≤ n is canonic as well as the restricted underlying matroids are canonic.

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We use the minimal lexicographic representation of the base function as canonic form for the underlying matroids. A chirotope is canonic by deﬁnition, if the underlying matroid is in canonic form, and if the orientations are lexicographically maximally distributed among all isomorphic chirotopes with canonic underlying matroid. This deﬁnition meets the described requirements for orderly generation and thus allows to do the denoted early canonicity tests during backtrack generation. During the distribution of orientations (second main level), the following additional tests are noteworthy: • When generating up to negation, we can restrict the ﬁrst otherwise freely choosable base orientation to +1. This is a necessary condition for the canonic form up to negation. • Similarly in the case when we generate up to reorientation: We ﬁx up to n orientations to +1. The chosen k-tuples are the ﬁrst ones aﬀected by an reorientation of the points 1 ≤ i ≤ n, respectively. (These tuples are computed once for each underlying matroid.) • Assure, that we generate chirotopes only: For all ordered (k + 2)-tuples (i) (i) (i) (x, y, a1 , . . . , ak ) with x < y < a1 , test the corresponding three-term Grassmann-Pl¨ ucker relations (GP3 ). Example 4. In Table 2 we demonstrate the splitting of the generation process into two main levels with the example of all simple oriented matroids of rank 4 over 7 points. On main level one, the listed 49 simple matroids are generated. For each of them, on main level two, all oriented matroids with given underlying matroid are generated. Remark that the main part of computation time is needed for generating the uniform oriented matroids. This is because we need to consider the whole S7 as operating group, thus the canonicity tests are quite expensive. For the other candidates of underlying matroids, we can restrict to smaller operating groups on the second main level, such that canonicity tests get easier. Also note that there are non-orientable matroids, i.e. matroids where there exists no corresponding oriented matroid. 3.2 Comparison Using origen, we recomputed published results of L. Finschi from [10]. We approved the information given there in Tables 6.3 and 6.4 in page 141 concerning isomorphism classes of simple oriented matroids up to relabeling, reorientation and negation as well as Tables 7.1 and 7.2 in page 151 concerning simple acyclic oriented matroids up to relabeling and negation alias abstract unsigned aﬃne point conﬁgurations (including degenerated cases). In Table 6.6 in page 143, Finschi gives some computation times for the generated structures of Table 6.3. We compared them with the computation times of origen. Note that origen is designed for handling several speciﬁc

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Table 2. Generation of simple oriented matroids of rank 4 over 7 points (up to relabeling, negation and reorientation) 12 1234 1235 1345 2345 1245 1236 1346 2346 1246 1356 2356 1456 2456 3456 1256 1237 1347 2347 1247 1357 2357 1457 2457 3457 1257 1367 2367 1467 2467 3467 1567 2567 3567 4567 67

Underlying matroid

××××××××××××××××××××××××××××××××××× 0 ×××××××××××××××××××××××××××××××××× 0 ×××××××××××××××××××××××××××××× 0 ××× 0 ×××××××× 0 ××××××××××××××××××××××××× 0 ×××××××× 0 ×××××××××××××× 0 ×××××××××× 0 ×××××××× 0 ×××××××××× 0 ×××××××××××××× 0 ×××××××× 0 ×××××××××× 0 ×××××××× 0 ××××× 0 ×××××××× 0 ×××××××××× 0 ××××××× 0 ×××××× 0 ×××××××× 0 ×××× 0 ×××××××××××××××××××× 0 ×××××××× 0 ×××× 0 ××××× 0 ×××××××××××××× 0 ×××××××× 0 ×××× 0 ××××× 0 ×××××××× 0 ××××× 0 ×××××××× 0 ×××× 0 ××××× 0 ×× 0 ××××××××××× 0 ×××××××× 0 ×××× 0 ××××× 0 ×× 0 ××× 0 ××××××× 0 ×××××××× 0 ×××× 0 ××××× 0 ×× 0 ××× 0 0 ×××××× 0 0 ××× 0 ××××××××× 0 ××××××××××××××××××× 0 0 ××× 0 ××××××××× 0 ×××××××××××××××××× 0 0 0 ××× 0 ×××××× 0 ×× 0 ××××××××××××××××××× 0 0 ××× 0 ×××××× 0 ×× 0 ××××××× 0 ××××××××××× 0 0 ××× 0 ×××××× 0 ×× 0 ××××××× 0 ×××××× 0 ×××× 0 0 ××× 0 ×××××× 0 0 0 0 ×××××××××××××××××× 0 0 0 0 0 0 ×××××××××××××××××××××××××××××× 0 0 0 0 0 ×××××××××××××××××××× 0 ××××××××× 0 0 0 0 0 ×××××××××××××××××××× 0 ×××× 0 ×××× 0 0 0 0 0 ×××××××××××××××××××× 0 0 × 0 ×× 0 ××× 0 0 0 0 0 0 ××××××××× 0 ××××××××××××××××××× 0 0 0 0 0 0 ××××××××× 0 ×××××××××××××××××× 0 0 0 0 0 0 0 ××××××××× 0 ×××××××××××× 0 ×××××× 0 0 0 0 0 0 ××××××××× 0 ×××××××××××× 0 ××× 0 ×× 0 0 0 0 0 0 ××××××××× 0 ×××××××××××× 0 0 0 ××× 0 0 0 0 0 0 0 ××××××××× 0 ××××××××× 0 0 0 ××××××× 0 0 0 0 0 0 ××××××××× 0 ××××××××× 0 0 0 ×××××× 0 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×××××××××××× 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×××××× 0 ××××× 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×××××× 0 ××× 0 × 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×× 0 0 0 ××××××× 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×× 0 0 0 × 0 ×× 0 ×× 0 0 0 0 0 0 ×××××× 0 ×× 0 ×××××× 0 ×× 0 0 0 0 ×× 0 ×× 0 0 0 0 0 0 0 0 0 0 ×××××× 0 0 0 0 ×××××××××××××××× 0 0 0 0 0 0 0 0 0 ×××××× 0 0 0 0 ×××××××××××× 0 ××× 0 0 0 0 0 0 0 0 0 ×××××× 0 0 0 0 ×××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××××××××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ××××××××××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××××××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××× 0 ×××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××× 0 ×××××× 0 ××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××× 0 ×××××× 0 ××× 0 × 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××××××××××××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ×××××××××××× 0 ××× 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ××××××××××

Group Orient-Computing order ed mat- time roids 5040 11 0.676 s 144 27 0.084 s 72 7 0.048 s 16 35 0.068 s 8 9 0.020 s 6 14 0.024 s 4 4 0.024 s 24 2 0.016 s 48 6 0.032 s 6 5 0.024 s 12 1 0.016 s 8 1 0.016 s 24 1 0.016 s 168 0 0.008 s 144 4 0.024 s 144 1 0.020 s 12 6 0.016 s 4 4 0.016 s 6 1 0.012 s 72 1 0.008 s 240 6 0.056 s 24 6 0.036 s 16 3 0.056 s 48 1 0.016 s 24 6 0.020 s 24 2 0.020 s 4 6 0.024 s 4 2 0.024 s 12 1 0.012 s 48 2 0.008 s 48 1 0.008 s 16 3 0.012 s 4 2 0.016 s 8 1 0.012 s 8 2 0.008 s 8 1 0.012 s 48 1 0.004 s 144 1 0.020 s 36 1 0.016 s 144 1 0.012 s 720 4 0.040 s 36 3 0.016 s 72 1 0.012 s 8 3 0.024 s 6 2 0.032 s 24 1 0.020 s 48 1 0.008 s 12 1 0.020 s 240 1 0.008 s Sum: 206

parameters as for example an acting symmetry group G which may diﬀer from Sn . This ensues a lot of overhead in implementation, which is not needed for the generation tasks in question. On the other hand, as our runs were about six years later than Finschi’s, our system (a Dual Core AMD Opteron (tm)

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Table 3. Number of isomorphism classes of oriented matroids. The entry for k = 7, n = 10 is new vs. Table 6.3 in [10] n= k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k = 10

1 1

2

3

4

5

6

7

8

9

10

1

1 1

1 2 1

1 4 3 1

1 17 12 4 1

1 143 206 25 5 1

1 4890 181 472 6 029 50 6 1

1 461 053

1 95 052 532

508 321 91 7 1

99 875 033 164 8 1

Table 4. CPU times needed for computing the results from Table 3 n= k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k = 10

Finschi’s generator 7 8 9 – 3s 10 s 2s – –

– 2.2 min 4.1 h 48.3 min 26 s – –

10

– 3.6 h

– ≈1700 h

≈240 h 9.9 min – –

origen 7 8 – 0.6 s 0.6 s∗ 0 s∗ – –

4.8 h – –

– 13 s 17 min 14 s∗ 0.2 s∗ – –

9

10

– 13 min

– 33 h

13.8 min∗ 0.6 s∗ – –

36 h∗ 1.5 s∗ – –

Table 5. Relative time speedups for k = 3 n= Ratio of comp. times

7 5

8 10

9 17

10 52

Processor 265, 1000 MHz, running on a 64 bit system) is most probably much faster than the one Finschi used (a Sun Sparc Ultra-60, 360 MHz). Thus only relative comparisons of CPU times do make sense. In Table 3, we list for given rank k and number n of points the number of simple oriented matroids up to reorientation, negation and relabeling, as given in Table 6.3 in [10], and as approved by origen. We added one number for the parameter set k = 7 and n = 10, which we were able to compute in less than two days. The CPU times given by Finschi as well as the ones needed by origen are shown in Table 4. For the sake of fairness, we need to mention, that we used a well known trick to compute some of the parameter sets: In the cases marked with a∗ , it was easier to compute the dual cosimple oriented

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matroids of rank k = n − k instead of direct computation. Finschi noted, that this is possible, but he did not make proﬁt of this trick. We made a relative comparison of computation times for k = 3, as there are four comparable times available in one row. In Table 5, we show the factor computation time by Finschi’s generator computation time by origen indicating a relative increase of computation speed with growing number n of points. Klin et al. computed numbers of non-isomorphic binary (i.e. non-degenerated) signed abstract 2D-conﬁgurations of n points, see Table 6 in [17]. The listed numbers for n = 3, 4, 5, 6 and 7 points, namely 1, 2, 3, 20 and 242 2D-conﬁgurations, respectively, coincide with the numbers of relabeling classes of acyclic uniform chirotopes of rank k = 3, as calculated by origen. However, a discrepancy for n = 8 remains, where Klin et al. reported 6406 abstract 2D-conﬁgurations, whereas origen constructed 6405 relabeling classes of acyclic uniform chirotopes.

4 Application in Chemical Conformation Analysis: The Example Cyclohexane Reconsider the chemical compound cyclohexane C6 H12 . We number the atoms as indicated in Fig. 5. We generate chirotopes of rank k = 4 over n = 6 points. The aﬃnely realizable ones thereof correspond to the three-dimensional point conﬁgurations of 6 atoms.

12 3 12 4 3 12 5 4 13 5 4 23 5 4 12 5 3 12 6 4 13 6 4 23 6 4 12 6 5 13 6 5 23 6 5 14 6 5 24 6 5 34 6 56

• The molecular graph has as automorphism group the dihedral group D6 with 12 elements. Thus, we generate chirotopes up to relabeling under the D6 . • Assuming that any 4 atoms are aﬃnely independent, we restrict to uniform chirotopes. Note that this is not a great restriction, as one can enforce general position of the atoms by inﬁnitesimal small movements. • Generating all isomorphism classes of uniform chirotopes over 6 points with D6 as acting group results in 386 solutions. The ﬁrst 5 are listed below: +++++++++++++++ ++++++++++++++– +++++++++++++– – ++++++++++++– ++ ++++++++++++– – + .. .

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Fig. 8. Two kinds of forbidden oriented circuits for molecular conformations (using bond-information from the molecular graph)

• We prescribe a list of forbidden circuits reﬂecting mathematical and chemical knowledge. – positive circuits are not aﬃnely realizable. – Three consecutively connected atoms have a triangle as convex hull. In general, we can exclude conformations where a bond intersects with such a triangle, as the steric energy of this conformation would be very high. Thus we can exclude all chirotopes containing a circuit of the form (+a, +b, +c, −d, −e) where atoms a, b, and c are consecutively connected as well as atoms d and e, see Fig. 8(a). – Similarly, we can exclude chirotopes having a circuit of the form (+a, +b, +c, +d, −e), where atom a is connected to atoms b, c and d, see Fig. 8(b). By excluding the forbidden circuits (+1, −3, −4, −5, +6), (+1, −2, −3, +5, +6),

(+1, −3, −4, +5, +6), (+1, +2, −4, −5, −6),

(+1, −2, −3, −4, +6), (+1, +2, −4, −5, +6),

(+1, +2, −3, −4, −5), (+1, +2, +3, −4, −5),

(+1, +2, −3, −4, +6), (+2, +3, −4, −5, −6),

(+1, +2, +3, −5, −6), (+2, +3, +4, −5, −6),

we reduce the number of solutions to 162. • Focusing on a set R of relevant quadruples of atoms, we consider partial chirotopes only. For example we could take as R the following quadruples of atoms: – the four neighbors of each stereocenter; These quadruples classify the stereoisomers; – four consecutively connected atoms. Using the relevant quadruples R = {1234, 2345, 1236, 1256, 1456, 3456}, we obtain 13 partial chirotopes shown in Table 6. Note that thus far we only considered combinatorial aspects. The remaining part is of geometrical nature, and there exist no sophisticated algorithms adjusted to the chemical situation, yet. Nevertheless using constrained optimization with the aid of the software package sqp by Gerdts [12], we can demonstrate, that in principle, the expected conformations are generated by the promoted twofolded strategy:

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+ + + + − − − + + + − − −

+ + − − + + − − + + + − +

34 56

24 56

14 56

23 56

13 56

12 56

23 46

+ + + + + + + − + + + + +

13 46

+ + + + + + + + − − − − −

12 46

12 36

13 45

12 45

23 45

+ + + + + + + + + + + + −

12 35

12 34

Table 6. Partial chirotopes generated for cyclohexane

+ − + − + − + + + − − − −

Fig. 9. Four conformations found by constrained optimization Table 7. Number of partial chirotopes generated for medium sized cyclic hydrocarbons Structure Partial chirotopes Comp. time

C6 H12 13 0s

C7 H14 18 0s

C8 H16 30 0.2 s

C9 H18 46 4.9 s

C10 H20 78 62 s

• For only 4 of the given partial chirotopes, we found a stable realization, see Fig. 9. Note that the ﬁrst of the conformations, known as boat form, is indeed an unstable conformation having zero gradient (i.e. a saddle point of the energy function). Though it is interesting that this was found by our algorithms, its unstability can easily be recognized by considering the Hesse matrix of the energy function. The remaining three conformations represent exactly the usual classiﬁcation of all possible conformations of cyclohexane (compare Fig. 9 with Fig. 2), namely the chair form (the third conformation) as well as two twisted forms (second and fourth conformation). Repeating the generation up to negation leads to 3 orientation patterns, reﬂecting the fact, that two of the found conformations (the two twisted forms) are mirror images of each other.

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We extended the combinatorial part of our example to medium-sized molecular structures, too. Table 7 shows numbers of partial chirotopes generated correspondingly for cyclic hydrocarbons with 6 to 10 carbon atoms together with the computation times needed by origen.

5 Conclusions and Outlook We presented origen, a generator of oriented matroids. Generation of oriented matroids can be seen as a ﬁrst combinatorial step towards the generation of conformations for chemical compounds. The second step, ﬁnding chemically feasible aﬃne realizations of the generated oriented matroids, is still an open task. There exist general algorithms ﬁnding aﬃne realizations of oriented matroids for small sizes [30, 28]. However the found realizations are often far away from being chemically feasible conformations. For more sophisticated approaches meeting the chemical demands, distances and maybe angles of the atoms should be taken into consideration. Thus distance geometry [6] as well as the Gr¨ obner bases approach by P. Hazebroek and L. Oosterhoﬀ [15] could be of great beneﬁt, here. A combination with the Gr¨ obner base approach could work out as follows: In Sect. 1 we shortly sketched how Hazebroek and Oosterhoﬀ formulated the structural properties of cyclohexane as polynomial equalities. It should not be a big problem to automatically build up systems of polynomial equations for arbitrary chemical structures, using tabularized standard bond lengths and bond angles for various bond and atom types. The prescribed orientations of a given partial chirotope actually are signs of determinants and thus directly lead to polynomial inequalities. Thus we can analyze an extended system of polynomial equations and inequalities corresponding to a conformation space with prescribed stereo-information. Depending on how much stereoinformation is prescribed, the solution space could be manageable small. Still, the open question is how diﬃcult it will be to solve the resulting systems of polynomial equations and inequalities.

Acknowledgments This work has been conducted during the Special Semester on Gr¨ obner Bases, February 1 – July 31, 2006, organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz, Austria. We especially want to thank B. Buchberger for creating this nice opportunity for discussion and further for the possibility of presentation in form of this paper. Special thank goes to M. Klin for detailed and motivating discussions.

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References 1. O. Aichholzer, F. Aurenhammer, and H. Krasser, Enumerating order types for small point sets with applications, Order, 19 (2002), 265–281. 2. O. Aichholzer and H. Krasser, The point set order type data base: A collection of applications and results, in Proc. 13th Annual Canadian Conference on Computational Geometry CCCG 2001, pp. 17–20, 2001. 3. A. Bj¨orner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids, 2nd edn., Cambridge University Press, Cambridge, 2000. 4. J. Bokowski, Computational Oriented Matroids, Cambridge University Press, Cambridge, 2006. 5. J. Bokowski and A. Guedes de Oliveira, On the generation of oriented matroids, Discrete Comput. Geom., 24 (2000), 197–208. 6. G. M. Crippen and T. F. Havel, Distance Geometry and Molecular Conformation, Research Studies Press, Taunton, 1988. 7. A. Dreiding and K. Wirth, The multiplex a classiﬁcation of ﬁnite ordered point sets in oriented d-dimensional space, Commun. Math. Comput. Chem., 8 (1980), 341–352. 8. A. Dress, A. Dreiding, and H. Haegi, Classiﬁcation of mobile molecules by category theory, Stud. Phys. Theor. Chem., 23 (1983), 39–58. 9. I. A. Faradˇzev, Constructive enumeration of combinatorial objects, Probl`emes Combinatoires et Th´eorie des Graphes, 260 (1978), 131–135. Colloq. Internat. CNRS, University of Orsay, Orsay 1976. 10. L. Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, PhD thesis, ETH Z¨ urich, 2001. 11. L. Finschi and K. Fukuda, Generation of oriented matroids, Discrete Comput. Geom., 27 (2002), 117–136. 12. M. Gerdts, SQP V1.1 – Ein Fortran77-Programmpaket zur L¨osung von nichtlinearen, restringierten Optimierungsproblemen, Universit¨ at Bayreuth, Bayreuth, 2004. 13. J. E. Goodman and R. Pollack, The complexity of point conﬁgurations, Discrete Appl. Math. (2), 31 (1991), 167–180. 14. R. Gugisch, Konstruktion von Isomorphieklassen orientierter Matroide, PhD thesis, University of Bayreuth, 2005. 15. P. Hazebroek and J. Oosterhoﬀ, The isomers of cyclohexane, Discrete Faraday Soc., 10 (1951), 87–93. 16. A. Heck, Introduction to MAPLE, 3rd edition, Springer, New York, 2003. 17. M. H. Klin, S. S. Tratch, and N. S. Zeﬁrov, 2D-conﬁgurations and cliquecyclic orientations of the graphs L(Kp ), Rep. Mol. Theory, 1 (1990), 149– 163. 18. D. E. Knuth, Axioms and Hulls, Lecture Notes in Computer Science, Vol. 606, Springer, Berlin, 1992. 19. R. Laue, Construction of combinatorial objects – a tutorial, Bayreuther Mathematische Schriften, 43 (1993), 53–96.

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Algorithmic Approach to Non-symmetric 3-class Association Schemes Leif K. Jørgensen Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7, 9220 Aalborg, Denmark. [email protected]

Summary. There are 24 feasible parameter sets for a primitive non-symmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove non-existence for three feasible parameter sets. Ten cases are still open. In the imprimitive case, we survey the known results including some constructions of inﬁnite families of schemes. In the smallest case that has been open up to now, we use computer search to ﬁnd new schemes. These schemes are equivalent to “skew” Bush-type Hadamard matrices of order 36. We also consider directed graphs that satisfy only some of the conditions required for a non-symmetric association scheme with 3 classes.

Key words: Association schemes, Orientation of strongly regular graphs, Computer search

1 Introduction The theory of association schemes was for a long time concentrated on the investigation of the symmetric association schemes generated by distance regular graphs. In this context the symmetric association schemes with two classes are exactly the schemes generated by strongly regular graphs. More opportunities appear as soon as we are dealing with at least three classes. A good survey of symmetric association schemes with three classes was provided by van Dam [6]. In this paper we consider non-symmetric association schemes with three classes. From each such association scheme, a symmetric association scheme with two classes can be obtained by merging the non-symmetric relations. Feasibility conditions for the existence of these association schemes have previously been considered by Bannai and Song [2], Song [37] and by Goldbach and Claasen [13]. In this paper we make an attempt of a more systematic investigation of non-symmetric 3-class association schemes with a relatively small number of M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 8,

251

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Leif K. Jørgensen

vertices. In the primitive case we generate all feasible parameter sets with at most 100 vertices. There are 24 such parameter sets. We review known results and prove non-existence results for three parameter sets, while 10 cases still remain open. We also brieﬂy consider normally regular digraphs (in the sense of [25]) as a generalization of non-symmetric 3-class association schemes. For the imprimitive case we start from a consideration of doubly regular (m, r)-team tournament in the sense of [27]. In [27] we distinguish three possible types of such directed graphs. A graph of type 3 cannot be a relation of an association scheme, however we do not know if any graph of this type exists. Types 1 and 2 indeed correspond to imprimitive non-symmetric 3-class association schemes. Graphs of type 1 are easily reduced to doubly regular tournaments in the sense of [35]. Thus we concentrate on graphs of type 2 and the corresponding association schemes. In particular we consider a subtype of type 2 which has links to Bush-type Hadamard matrices. Here we investigate the smallest open case of order 36. We ﬁnd four such association schemes by computer search, but we leave open the problem of complete enumeration of all association schemes with this set of parameters. We expect that there may be a large number of such schemes – probably all with small automorphism groups. N. Ito [19] has proved that they cannot have an automorphism group of rank 4. It should be stressed that our approach is strictly algorithmic, essentially depending on the use of computers. The computer is used already on the initial stage of the generation of all feasible sets of parameters of primitive schemes. For computer-aided constructive enumeration of all association schemes with a given set of parameters we use two diﬀerent approaches. The ﬁrst approach makes use of a complete catalog of strongly regular graphs with the parameters that would be obtained by merging the non-symmetric relations. In the cases considered in this paper we use the complete catalog of strongly regular graphs with parameters (45, 12, 3, 3) found by Coolsaet, Degraer and Spence [5] and the classical result of Hoﬀman and Singleton [16] about the uniqueness of the strongly regular graph with parameters (50, 7, 0, 1). The second approach is an orderly generation algorithm in the spirit of Faradˇzev [8] and Read [34]. These techniques are used to exclude existence of three feasible parameter sets for primitive association schemes. The second technique was also used to ﬁnd the above mentioned imprimitive association schemes of order 36. In that case the search space is huge and it was not possible to complete the full search. But we successfully used some ad hoc tricks in order to catch in the whole search space a few lucky directions leading to a construction of the desired combinatorial objects. We hope that the results presented in this paper may help to promote further approaches towards constructive enumeration of association schemes.

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2 Preliminaries Let X be a ﬁnite set (|X| = v) and let {R0 , R1 , . . . , Rd } be a partition of X × X. Then we say that X = (X, {R0 , R1 , . . . , Rd }) is an association scheme with d classes if the following conditions are satisﬁed • R0 = {(x, x) | x ∈ X}, • for each i, Rit := {(x, y) | (y, x) ∈ Ri } = Ri , for some i , • and for each triple (i, j, h), i, j, h ∈ {0, . . . , d} there exists a so-called intersection number phij such that for all x, y ∈ X with (x, y) ∈ Rh there are exactly phij elements z ∈ X so that (x, z) ∈ Ri and (z, y) ∈ Rj . For i > 0 the relation Ri can be viewed as the edge set of the (undirected or directed) graph (X, Ri ). We will frequently identify this graph with the relation Ri . If i = i for all i then X is said to be symmetric, otherwise it is nonsymmetric. If the graphs R1 , . . . , Rd all are connected then we say that X is primitive, otherwise it is imprimitive. A relation (say R1 ) of a symmetric association scheme with two classes is a strongly regular graph with parameters (v, k, a, c), where v = |X|, k = p011 , a = p111 , c = p211 . And conversely, if R1 is a strongly regular graph and R2 is the complementary graph of R1 , then R1 and R2 form a symmetric association scheme with two classes. A relation of a non-symmetric association scheme with two classes is called a doubly regular tournament. Reid and Brown [35] proved that there exists a doubly regular tournament with n vertices if and only if there exists a skew Hadamard matrix of order n + 1. Thus a necessary condition is that n ≡ 3 mod 4. In this paper we consider non-symmetric association schemes with d = 3 classes. We will assume that the relations are enumerated so that R1 and R2 are non-symmetric, R2 = R1t , and R3 is a symmetric relation. In this case the association scheme is determined uniquely by relation R1 . If A denotes the adjacency matrix of the relation R1 then the adjacency matrices of R0 , R2 and R3 are I, At and J −I −A−At , respectively. The BoseMesner algebra of X is the matrix algebra A spanned by these four matrices, see Bannai and Ito [1]. Higman [15] proved that an association scheme with d ≤ 4 has a commutative Bose-Mesner algebra, which means that phij = phji , for all i, j, h. Thus multiplication in the Bose-Mesner algebra is determined by the following equations. AJ = JA = κJ,

(1)

AA = κI + λ(A + A ) + μ(J − I − A − A ), At A = κI + λ(A + At ) + μ(J − I − A − At ), A2 = αA + βAt + γ(J − I − A − At ),

(2) (3) (4)

t

t

t

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where κ = p012 , λ = p112 = p121 = p212 = p221 , μ = p312 = p321 , α = p111 , β = p211 and γ = p311 . We note that α = λ. This is seen by counting in two ways the pairs (y, z) so that (x, y), (x, z), (y, z) ∈ R1 , for a ﬁxed vertex x. Since A is commutative and consists of normal matrices, the matrices of A have a common diagonalization, i.e., A has a basis {E0 , E1 , E2 , E3 } of orthogonal projections. Since a non-symmetric association scheme X with 3 classes is commutative, the symmetrization (X, {R0 , R1 ∪ R2 , R3 }) is also an association scheme, thus R3 is a strongly regular graph and R1 and R2 are orientations of a strongly regular graph. In fact R1 ∪ R2 is a strongly regular graph with parameters (v, k, a, c) = (v, 2p012 , p111 + p112 + p121 + p122 , p311 + p312 + p321 + p322 ) = (v, 2p012 , 3p112 + p122 , 2(p311 + p312 )).

(5)

In [25], we prove the following. Lemma 1. If A is the adjacency matrix of a regular directed graph (i.e., (1) is satisﬁed), then (2) and (3) are equivalent. (This is also an alternative proof of the commutativity of the Bose-Mesner algebra A in this particular case.) A directed graph whose adjacency matrix satisﬁes these equations is called a normally regular digraph. The eigenvalues of a normally regular digraph have the following property. Theorem 1. (See [25].) If the adjacency matrix A of a regular directed graph satisﬁes (2) then an eigenvalue θ = k lies on the circle in the complex plane with center λ − μ and radius k − μ + (λ − μ)2 and θ + θ is an eigenvalue of A + At . If A satisﬁes all Eqs. 1, 2, 3 and 4 then it has four eigenvalues κ, and say ρ, σ and σ with multiplicities 1, m1 , m2 and m2 , respectively, and the eigenvalues of A + At are 2κ, 2ρ, and σ + σ with multiplicities 1, m1 , and 2m2 . For parameters v and phij , i, j, h ∈ {0, 1, 2, 3} the parameters of R1 ∪ R2 can be computed from (5). Using standard formulas, the spectrum of R1 ∪ R2 can then be computed. From this it is possible to compute eigenvalues and multiplicities of R1 (e.g. using Theorem 1). For an arbitrary set of intersection numbers, using expressions for multiplicities, one may get non-integer values, thus excluding this parameter set. Deﬁnition 1. We say that v and phij , i, j, h ∈ {0, 1, 2, 3} form a feasible parameter set for a non-symmetric association scheme with three classes if they are non-negative integers and the multiplicities of the (four) eigenvalues computed from these intersection numbers are positive integers. Bannai and Song proved that the spectrum of A can be computed from the spectrum of A + At . (We note that if the eigenvalues of A + At are 2κ, r, s then either r or s can be split in two complex eigenvalues, if their multiplicities are even.)

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Lemma 2. (See Bannai and Song [2].) Suppose A is an adjacency matrix of a non-symmetric relation R1 of a 3-class association scheme. If s is the eigenvalue (of multiplicity 2m) of A + At that issplit in two complex eigenvalues σ and σ (i.e., s = σ + σ) then σ = 12 (s + i vκ/m). It is well known that the intersection numbers can be computed from the spectrum of A. The Hadamard product of matrices B = (bij ) and C = (cij ) is the matrix B ◦ C = (bij cij ). Since {I, A, At , J − A − At − I} is a basis of A, it follows by considering the Hadamard product of these matrices that A is closed under h , for i, j, h ∈ the Hadamard product. In particular there exist numbers qij h 1 {0, 1, 2, 3}, so that Ei ◦ Ej = v h qij Eh . These numbers are called Krein parameters. It is known that each Krein parameter is a non-negative real number, see Bannai and Ito [1]. Since the Krein parameters can be computed from the spectrum of A, this can be used to prove non-existence for some feasible parameter sets. Neumaier [33] found another way to exclude feasible parameter sets. Let mi be the rank of Ei , for i ∈ {0, 1, 2, 3}. (Thus m0 , . . . , m3 are the multiplicities of eigenvalues.) Theorem 2. (See [33].) The following inequalities are satisﬁed for a commutative association scheme. 1 mh ≤ mi (mi + 1), for i = 0, . . . , d, 2 h >0 h:qii mh ≤ mi mj , for i, j = 0, . . . , d, i = j. h >0 h:qij

3 Primitive Association Schemes with Three Classes We now use a computer to generate a list of all feasible parameter sets for primitive association schemes with three classes and |X| ≤ 100. For each feasible parameter set (v, k, a, c) of a strongly regular graph we investigate the feasible parameters of non-symmetric association schemes with three classes such that R1 ∪ R2 has parameters (v, k, a, c). It follows from (5) that we need only consider parameters where k and c are even. It is also useful to know that the eigenvalues of R1 ∪ R2 are integers. This follows from the next lemma. Lemma 3. (See Goldbach and Claasen [13].) There is no non-symmetric association scheme with three classes so that R1 ∪ R2 has parameters (4c + 1, 2c, c − 1, c). In Goldbach and Claasens’s terminology they proved non-existence if the strongly regular graph is pseudo-cyclic, i.e., the two non-trivial eigenvalues have the same multiplicities. It is well-known that this is equivalent to having

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Table 1. A list of all feasible parameter sets for primitive non-symmetric 3-class association schemes with at most 100 vertices. The second column is the parameters of the strongly regular graph R1 ∪ R2 . The third column gives information on the number of strongly regular graphs with these parameters. (In fact in some cases where we write ≥ 1, there are several known strongly regular graphs, e.g. with parameters (100, 44, 18, 20), see [26].) These numbers are from [3, 5, 32]. In column six “NO” means that we prove non-existence of the association scheme in this paper and “no” means that non-existence follows from general results or it was proved in other papers No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Parameters for R1 ∪ R2 (16, 10, 6, 6) (21, 10, 3, 6) (36, 14, 4, 6) (36, 20, 10, 12) (45, 32, 22, 24) (50, 42, 35, 36) (57, 42, 31, 30) (64, 28, 12, 12) (64, 36, 20, 20) (64, 42, 26, 30) (64, 42, 30, 22) (81, 50, 31, 30) (85, 64, 48, 48) (85, 70, 57, 60) (96, 38, 10, 18) (96, 50, 22, 30) (96, 60, 38, 36) (96, 76, 60, 60) (100, 44, 18, 20) (100, 54, 28, 30) (100, 66, 39, 52) (100, 66, 41, 48) (100, 66, 44, 42) (100, 72, 50, 56)

No. of SRGs 1 1 180 32548 78 1 0 ≥1 ≥1 ≥1 0 ≥1 ≥1 ? ? ? ? ≥1 ≥1 ≥1 0 ≥1 ? ≥1

p112

p312

Exists

Reference

1 1 0 3 6 8 7 4 4 7 7 9 13 13 3 3 11 16 3 8 10 8 10 13

2 1 2 2 4 12 9 2 6 6 6 5 8 20 4 10 6 10 6 6 12 16 12 12

no no yes NO NO NO no yes yes ? no ? ? ? ? no no ? ? ? no no ? ?

Goldbach and Claasen [12] Enomoto and Mena [7] Goldbach and Claasen [11] Theorem 5 Theorem 4 Theorem 3 Wilbrink and Brouwer [38] Enomoto and Mena [7] Jørgensen [28] Absolute bound

Neumaier Krein

Absolute bound Neumaier

parameters (4c + 1, 2c, c − 1, c), see [3]. (Only in this pseudo-cyclic case it is possible that a strongly regular graph has irrational eigenvalues.) The resulting list of feasible parameter sets is presented in Table 1. The association scheme with parameter set no. 3 was constructed by Iwasaki [23] and independently by Ivanov, Klin and Faradˇzev [22], see also [9]. Later Goldbach and Claasen [11] proved that it is the unique association scheme with these parameters. The association scheme with parameter set no. 8 was constructed by Enomoto and Mena [7]. Liebler and Mena [30] showed this scheme belongs to an inﬁnite family of association schemes. These

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schemes have order 4s4 where s is a power of 2. Four association schemes with parameter set no. 9 were constructed in [28]. These are the only known primitive non-symmetric association schemes with three classes. In parameter sets nos. 7, 11 and 21 it is known that the strongly regular graph does not exist, see Brouwer [3]. Thus the 3-class association scheme does not exist in these cases. In parameter set no. 17 some of the Krein parameters are negative. Thus this case is excluded. The multiplicities of eigenvalues for parameter sets nos. 16 and 22 do not satisfy Neumaier’s condition. We will now use computers to prove non-existence of association schemes with parameter sets nos. 4, 5 and 6. We use two diﬀerent techniques. For parameter sets nos. 5 and 6, the computation is based on the classiﬁcation of all strongly regular graphs which have the same parameters as the strongly regular graph R3 (assuming that (X, {R0 , R1 , R2 , R3 }) is the required association scheme). For a given graph R3 we try to construct R1 by orienting the complement of R3 . We ﬁrst consider orientation of edges of the complement of R3 incident with a ﬁxed vertex x. We let N + (x) denote the outneighbors of x in R1 and we let N2 (x) denote the vertices at distance 2 from x in R3 . Then a candidate for N + (x) consists of exactly half of the vertices in N2 (x). But it also has some other properties. Let x1 , . . . , xk be the neighbors of x in R3 and Si denote the set of neighbors of xi in N2 (x), for i = 1, . . . , k. Then a candidate for N + (x) must satisfy |Si ∩ N + (x)| = p313 = 12 |Si |. It also satisﬁes that the subgraph of R3 induced by N + (x) is regular of degree p113 and the subgraph induced by N2 (x) \ N + (x) is regular of degree p223 = p113 . When we have computed the list of all candidates for N + (x) for every vertex x, we try if it is possible to combine these orientations in such a way that for any two vertices x and y the orientation of the edges incident with x and the orientation of the edges incident with y should agree on the orientation of the edge {x, y} if x and y are non-adjacent in R3 , and they should satisfy that for all i, j the number of vertices z so that (x, z) ∈ Ri and (z, y) ∈ Rj is exactly phij where (x, y) ∈ Rh . For parameters no. 6, R3 is a strongly regular graph with parameters (50, 7, 0, 1), i.e., it is the Hoﬀman-Singleton graph, see [16]. This case can be excluded by investigating possible orientations of the complement of the Hoﬀman-Singleton graph. Theorem 3. There is no non-symmetric association scheme with three classes where R3 is the Hoﬀman-Singleton graph. Proof. Suppose that there exists a non-symmetric association scheme with three classes where R3 is the Hoﬀman-Singleton graph. When applying the method described above we may use that the Hoﬀman-Singleton graph has a large group of automorphisms. Computations using this group are done in GAP [10] with GRAPE [36] and nauty [31]. Other computations are done in a C-program.

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Let x be a vertex and let x1 , . . . , x7 be the neighbors of x in R3 . Let Si be the set of neighbors of xi other than x, for i = 1, . . . , 7. Let N + (x) be the set out-neighbors of x in R1 . Then N + (x) is a set of 21 vertices in the set N2 (x) = S1 ∪ . . . ∪ S7 of vertices at distance 2 from x, and |Si ∩ N + (x)| = p313 = 3, for i = 1, . . . , 7. The subgraph of R3 spanned by N + (x) is regular of degree p113 = 4. The complement of N + (x) in S1 ∪ . . . ∪ S7 is the set of in-neighbors of x in R1 and this set also spans a 4-regular subgraph of R3 . A computer enumeration shows that there are exactly 1140 subsets of N2 (x) with the properties required for N + (x). These 1140 subsets form three orbits under the action of the subgroup of the automorphism group of the Hoﬀman-Singleton graph stabilizing the vertex x. Thus we need only consider three possibilities for N + (x), but then we must consider all 1140 candidates N + (y) for any other vertex y. It turns out that we only need to consider orientations of edges incident with x, x1 , . . . , x5 . These edges must by oriented such that |N + (x) ∩ N + (xi )| = p312 = 12, as / R3 , and such (x, xi ) ∈ R3 , |N + (xi ) ∩ N + (xj )| = p112 = p212 = 8, as (xi , xj ) ∈ / N + (xj ). that xj ∈ N + (xi ) if and only if xi ∈ A computer search shows that there are no orientations of all edges incident with x, x1 , x2 , x3 , x4 and x5 that satisfy these conditions. Thus the required association scheme does not exist.

For parameters no. 5, R3 is a strongly regular graph with parameters (v, k, a, c) = (45, 12, 3, 3). Coolsaet, Degraer and Spence [5], have shown that there are exactly 78 strongly regular graphs with these parameters. Thus the method from the previous theorem can be applied to each of these 78 graphs. Theorem 4. There is no primitive non-symmetric association scheme with three classes with parameter set no. 5. Proof. Suppose that there exists such an association scheme. Let x be a vertex and let x1 , . . . , x12 be the neighbors of x in R3 . Let Si be the set of neighbors of xi at distance 2 from x, |Si | = k − a − 1 = 8, for i = 1, . . . , 12. Let N + (x) be the set out-neighbors of x in R1 . Then N + (x) is a set of 16 vertices in the set N2 (x) := S1 ∪ . . . ∪ S12 , and |Si ∩ N + (x)| = p313 = 4, for i = 1, . . . , 12. The subgraph of R3 spanned by N + (x) is regular of degree p113 = 3. The computer search shows that if N is a set with |Si ∩ N | = 4, for i = 1, . . . , 12, and in which every vertex has degree at most 3 then N is 3-regular and the subgraph of R3 spanned by N2 (x) \ N is also 3-regular. The number of such sets N depend on the graph and the vertex x. The largest number of sets is 396, which appear in the graph with a rank 3 automorphism group. 44 of the 78 candidates for R3 can be excluded because, for at least one vertex x, there is no such set N . For each of the other 34 graphs we ﬁnd by computer search a set W of at most 8 vertices so that there is no combination of orientations of edges in the

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complement of R3 incident with w, for each w ∈ W that satisﬁes the required properties. (This search took 45 minutes on a 2.4 GHz PC.) Thus an association scheme with parameter set no. 5 does not exist.

For parameter set no. 4 (and for one case of imprimitive association schemes, see Sect. 4) we use a diﬀerent computer search technique. This does not depend on characterization of strongly regular graphs. We use an orderly generation algorithm (see Faradˇzev [8] or Read [34]) to search for the matrix B = 3A3 + 2A2 + A1 , where A1 , A2 , A3 are adjacency matrices of the relations R1 , R2 , R3 of the required association scheme. Recall that for i ∈ {1, 2, 3} we deﬁne i ∈ {1, 2, 3} so that Rit = Ri . In our usual enumeration of relations this means that 1 = 2, 2 = 1 and 3 = 3, but in the ﬁrst application of the algorithm (Theorem 5) we use a diﬀerent enumeration (where 1 = 1, 2 = 3 and 3 = 2). We want the vertices to be labeled with numbers 1, . . . , v so that the matrix B is in maximal form, i.e., the sequence obtained by reading the entries of the ﬁrst row followed by the entries of the second row, etc., is as large as possible (in the lexicographic order) among all labellings of the vertices. Suppose that the ﬁrst r − 1 rows of the matrix B = (bij ) has been ﬁlled in. We then investigate all possible ways to ﬁll in row r with 0 on the diagonal entry, p011 entries with 1’s, p022 entries with 2’s, and p033 entries with 3’s in such a way that • the ﬁrst r − 1 entries are in accordance with the entries of column r of the previous rows, • for each x < r the number of columns s, so that bxs = i and brs = j is exactly phij , where bxr = h, • the matrix is still in maximal form. For each possible way to ﬁll row r we repeat the procedure for row r + 1. Theorem 5. There is no primitive non-symmetric association scheme with three classes with parameter set no. 4. Proof. As described above, we search for the matrix B = 3A3 + 2A2 + A1 . In turns out that with the maximality condition on the matrix and for this particular parameter set it is convenient to enumerate the relations so that R1 is symmetric and R2t = R3 . Thus the ﬁrst row of B should consist of one 0 followed by p033 = 10 entries with 3’s followed by p022 = 10 entries with 2’s and ﬁnally p011 = 15 entries with 1’s. When using the algorithm described above we ﬁnd that the number of ways to ﬁll in the ﬁrst r rows is 1, 1, 100, 24161, 205671, 1116571, 52650, 39, 0, . . . , 0, for r = 1, . . . , 36. Thus the required association scheme does not exist. (This search took 81 minutes on a 2.4 GHz PC.)

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4 Imprimitive Association Schemes with Three Classes 4.1 General Results If R3 is connected but R1 and R2 are disconnected then each connected component of R1 is a doubly regular tournament on 2p012 + 1 vertices. Thus the study of these schemes reduces to the study of doubly regular tournaments. We will thus assume that R1 and R2 are connected and R3 is disconnected. Then R3 consists of m copies of a complete graph on r vertices, for some constants m and r. We denote this graph by m◦Kr . Then R1 is an orientation of the complement m ◦ Kr . The vertex set of m ◦ Kr is partitioned in m independent sets of size r, denoted by V1 , . . . , Vm . In [27] we introduce the following family of graphs that do not necessarily satisfy all the conditions on a relation of a non-symmetric association scheme with three classes. We say that a directed graph is a doubly regular (m, r)team tournament if it is an orientation of m ◦ Kr with adjacency matrix A satisfying (1) and (4) in Sect. 2. In [27] we give a combinatorial proof of the following, i.e., we do not use eigenvalues. Theorem 6. (See Jørgensen, Jones, Klin and Song [27].) Every doubly regular (m, r)-team tournament is of one of the following types. 1. For every pair i, j either all the edges between Vi and Vj are directed from Vi to Vj , or they are all directed from Vj to Vi . The graph with vertices v1 , . . . , vm and an edge directed from vi to vj if edges are directed from Vi to Vj is a doubly regular tournament. 2. For every vertex x ∈ Vi , exactly half of the vertices in Vj (j = i) are 2 , and γ = (m−1)r out-neighbors of x, and α = β = (m−2)r 4 4(r−1) . 3. For every pair {i, j} either Vi is partitioned in two sets Vi and Vi of size 2r so that all edges between Vi and Vj are directed from Vi to Vj and from Vj to Vi , or similarly with i and j interchanged. The parameters are 2 (m−1)r − 3r + 8r , γ = (m−1)r α = (m−1)r 4 8 , β = 4 8(r−1) . A graph of type 3 cannot be a relation of an association scheme. In this case 8 divides r and 4(r − 1) divides m − 1. We do not know if any graph of this type exists. Every graph of type 1 or type 2 is a relation of a non-symmetric association scheme with 3 classes. The results for these types where ﬁrst proved by Goldbach and Claasen [14]. Clearly, the graph of type 1 exists if and only if a doubly regular tournament of order m exists. Thus in the remaining part this section we will only consider graphs of type 2.

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4.2 Association Schemes of Type 2 We ﬁrst show that a graph of type 2 is a relation of a non-symmetric association scheme with 3 classes. This is done by proving that (2) and (3) are satisﬁed. Lemma 4. Let A be the adjacency matrix of a doubly regular (m, r)-team tournament of type 2. Then A satisﬁes (2) and (3) with • λ = α = (m−2)r and 4 (m−1)r(r−2) • μ= . 4(r−1) In particular if m = r then λ = μ =

m(m−2) . 4

Proof. Let x ∈ Vi and y ∈ Vj , i = j, and suppose that there is an edge directed from x to y. Then x has κ − 2r out-neighbors outside Vi ∪ Vj , α of these are in-neighbors of y and the remaining κ − 2r − α are out-neighbors of y. Thus λ = κ − 2r − α = (m−2)r , since κ = (m−1)r . 4 2 Similarly, for x, y ∈ Vi , we get μ = κ − γ = (m−1)r(r−2) . Thus (2) is 4(r−1) satisﬁed. Equation 3 can be proved in a similar way, or by applying Lemma 1.

Since the parameters of a graph of type 2 are integers, it follows that r is even and r − 1 divides m − 1. Using eigenvalues, it can be shown that m is even, see [27] or Goldbach and Claasen [14]. Existence in the case r = 2 is equivalent to existence of a doubly regular tournament of order m − 1. Theorem 7. (See [27].) If there exists a doubly regular (m, 2)-team tournament Γ then 4 divides m and the out-neighbors of a vertex in Γ span a doubly regular tournament of order m − 1. Conversely, for every doubly regular tournament T of order m−1, there exists a doubly regular (m, 2)-team tournament Γ , such that for some vertex x in Γ the out-neighbors of x span a subgraph isomorphic to T . In [28] we found 40 association schemes with (r, m) = (4, 16). No other schemes with 4 ≤ r < m, where r − 1 divides m − 1 are known. We will now consider the case m = r. By Lemma 4, the directed graph is then a normally regular digraph with μ = λ. Such digraphs are also known as doubly regular asymmetric digraphs. These graphs were introduced and studied in a series of papers by N. Ito [18–21] and also studied by Ionin and Kharaghani [17]. The ﬁrst non-trivial case of an association scheme of type 2 with m = r is for m = 4. In this case there exist two non-isomorphic schemes. One of these schemes has an automorphism group of rank 4, i.e., the group acts transitively on the vertices and the stabilizer of a vertex x has four orbits: {x}, the set of out-neighbors of x, the set of in-neighbors of x and the set of vertices not

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adjacent to x. Any doubly regular asymmetric digraph with automorphism group of rank 4 is a relation of a non-symmetric association scheme with 3 classes. Ito [19] has proved that a primitive non-symmetric 3-class association scheme with μ = λ does not satisfy the feasibility condition. Thus a doubly regular asymmetric digraph with automorphism group of rank 4 is a relation of an imprimitive non-symmetric 3-class association scheme of type 2 with m = r (as μ = λ). In this case Ito [19] has proved that m = r is a power of 2. He also claims to have proved that the only possibility is m = 4. But the proof of this does not seem to be correct and in fact Ito in his paper gives an example of a vertex transitive scheme with m = r = 8. According to computations in GAP [10] using share package GRAPE [36] with nauty [31] the automorphism group of this scheme has rank 4. We will now consider the links between such association schemes and a special case of some well-known structures. Deﬁnition 2. An Hadamard matrix H of order n is an n × n matrix in which every entry is either 1 or −1 and HH t = nI. An Hadamard matrix H of order m2 is said to be Bush-type if H is block matrix with m × m blocks Hij of size m × m such that Hii = Jm and Hij Jm = Jm Hij = 0, for i = j. Theorem 8. An imprimitive 3-class association scheme of type 2 and with r = m is equivalent to a Bush-type Hadamard matrix of order m2 with the t , for all pairs i, j with i = j. property that Hij = −Hji Proof. Let A be an adjacency matrix of relation R1 , for some imprimitive 3class association scheme of type 2 and with r = m. We may assume that vertices are enumerated such that the vertices in Vi corresponds to columns/rows mi − m + 1, . . . , mi. Let H = Jm2 − 2A. Then H is partitioned in blocks Hij of size m × m corresponding to the partition of vertices in sets V1 , . . . , Vm . Clearly Hii = Jm and since a vertex in Vi has exactly m 2 out-neighbors and m in-neighbors in V , H J = J H = 0. j ij m m ij 2 From (1) and (2) we get (since κ = m(m−1) and μ = λ = m(m−2) ) 2 4 HH t = (Jm2 − 2A)(Jm2 − 2At ) = (m2 − 4κ)Jm2 + 4(κI + μ(Jm2 − I)) = m2 I. Thus H is an Hadamard matrix. Conversely, suppose that H is a Bush-type Hadamard matrix which is t skew in the sense that Hij = −Hji , for i = j. 1 Let A = 2 (J − H), where J = Jm2 . Then A is a {0, 1} matrix. Since H m is Bush-type it has exactly m + (m − 1) m 2 entries equal to 1 and (m − 1) 2 entries equal to −1 in each row. Thus HJ = mJ and the transposed equation J and is JH t = mJ. Similarly JH = mJ. Thus AJ = JA = m(m−1) 2 AAt =

m2 1 m(m − 2) (J − H)(J − H t ) = J+ I. 4 4 4

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We see that (1) and (2) are satisﬁed. Equation 3 can be proved in a similar way, or by applying Lemma 1. Let K denote the block diagonal matrix with diagonal blocks equal to Jm . Then the Bush-type property of H implies that HK = mK and the skew property of H implies that H + H t = 2K. Thus H 2 = H(2K − H t ) = 2mK − m2 I, and so A2 =

1 1 (J − H)2 = (m(m − 2)J + 2mK − m2 I). 4 4

Since J −I−A−At = K−I, it follows that (4) is satisﬁed with α = β = 2 and γ = m4 .

m(m−2) 4

Kharaghani [29] proved that if there exists an Hadamard matrix of order m then there exists a Bush-type Hadamard matrix of order m2 . Ionin and Kharaghani [17] modiﬁed this construction and proved that if there exists an Hadamard matrix of order m then there exists a Bushtype Hadamard matrix of order m2 , which has the skew property required in Theorem 8. Thus in many cases with m = r a multiple of 4, an association scheme can be constructed. The case with m = r congruent to 2 modulo 4 seems to be more diﬃcult and no general constructions are known. But in the special case m = r = 6 we may apply the orderly generation algorithm described before Theorem 5. The number of ways to ﬁll the ﬁrst s rows is 1, 1, 4, 12, 8, 6, 29077, 76216458, for s = 1, 2, . . . , 8. (Note that the ﬁrst six rows correspond to a connected component of the undirected graph R3 .) We estimate that a complete search through all 76 million ways to ﬁll the ﬁrst 8 rows would take several years. But we guessed (especially because there are no such schemes with a rank 4 group) that if a scheme exists then there are many schemes and so a partial search may lead to a least one scheme. Probably starting a complete search and let the computer run until a scheme is discovered is not an optimal strategy. Instead, we chose 2405 cases randomly among all ways to ﬁll 8 rows. This search gave 47 ways to ﬁll 13 rows but no ways to ﬁll 14 rows. The idea is now to do a complete search in the “neighbourhood” of the most successful 8-row matrices, where the neighborhood of an 8-row matrix is the set of all 8-row matrices with which it has the ﬁrst 7 rows in common. This search lead to two association schemes. A repetition (with another set of randomly chosen 8-row matrices) gave two other schemes. Thus we have: Theorem 9. There exist at least four imprimitive non-symmetric 3-class association schemes of type 2 with m = r = 6. Each of these four schemes has a trivial automorphism group.

264

Leif K. Jørgensen Table 2. Matrix of a 3-scheme with m = r = 6

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 1 0 0 0

1 1 0 1 0 0

1 0 0 1 0 1

0 1 0 0 1 1

0 0 1 1 1 0

0 0 1 0 1 1

1 1 1 0 0 0

1 0 0 1 1 0

1 0 0 0 1 1

0 1 1 0 0 1

0 1 0 1 1 0

0 0 1 1 0 1

1 1 0 1 0 0

1 0 1 0 0 1

1 0 0 0 1 1

0 1 1 0 0 1

0 1 0 1 1 0

0 0 1 1 1 0

1 1 0 0 1 0

1 1 0 0 0 1

1 0 1 0 1 0

0 1 1 1 0 0

0 0 1 1 0 1

0 0 0 1 1 1

1 0 1 1 0 0

1 0 1 0 1 0

1 0 0 1 0 1

0 1 1 0 1 0

0 1 0 1 0 1

0 1 0 0 1 1

0 0 0 1 1 1

0 0 1 0 1 1

0 1 1 1 0 0

1 0 0 1 0 1

1 1 1 0 0 0

1 1 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 1 0 0 0 1

1 0 1 0 1 0

0 0 1 1 1 0

1 0 0 1 0 1

0 1 0 0 1 1

0 1 1 1 0 0

0 0 1 1 0 1

1 1 0 0 1 0

0 1 0 0 1 1

0 0 1 1 1 0

1 1 0 1 0 0

1 0 1 0 0 1

1 0 0 1 1 0

0 1 1 0 0 1

1 0 1 1 0 0

0 1 1 0 1 0

1 0 0 0 1 1

0 1 0 1 0 1

0 0 0 1 1 1

0 1 1 0 0 1

1 1 0 1 0 0

0 1 0 1 1 0

1 0 1 0 1 0

1 0 1 0 0 1

0 0 0 1 1 1

0 1 1 0 0 1

0 1 1 0 1 0

1 0 1 1 0 0

1 0 0 1 0 1

1 1 0 0 1 0

0 0 1 0 1 1

0 1 1 1 0 0

1 0 0 1 1 0

1 1 0 0 1 0

1 0 0 1 0 1

0 1 1 0 0 1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 1 1

1 0 1 0 0 1

0 0 1 1 1 0

1 1 0 1 0 0

0 1 1 1 0 0

0 1 0 0 1 1

1 1 0 0 0 1

0 1 0 1 0 1

0 0 1 1 1 0

1 0 1 0 1 0

0 1 0 1 1 0

1 0 1 0 0 1

1 1 1 0 0 0

1 1 0 1 0 0

0 1 0 0 1 1

0 0 0 1 1 1

0 0 1 1 0 1

1 0 1 0 1 0

0 0 0 1 1 1

0 1 1 0 0 1

1 0 1 0 1 0

0 1 1 1 0 0

1 1 0 1 0 0

1 0 0 0 1 1

1 0 1 1 0 0

1 0 0 1 0 1

0 1 1 0 1 0

0 1 1 0 0 1

1 0 0 0 1 1

0 1 0 1 1 0

0 0 1 0 1 1

1 1 1 0 0 0

1 0 0 1 0 1

1 1 0 0 0 1

0 1 0 1 1 0

0 0 1 1 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 1 1 1 0

1 1 0 0 1 0

0 1 0 0 1 1

0 1 0 1 0 1

1 0 1 1 0 0

1 0 1 0 0 1

1 1 0 0 1 0

0 0 1 1 0 1

0 1 0 1 0 1

1 1 1 0 0 0

0 0 1 0 1 1

1 0 0 1 1 0

0 0 0 1 1 1

0 0 1 0 1 1

1 1 0 0 0 1

1 1 1 0 0 0

0 1 0 1 1 0

1 0 1 1 0 0

0 1 0 1 0 1

1 0 1 0 1 0

1 0 0 0 1 1

0 1 0 1 1 0

0 1 1 0 0 1

1 0 1 1 0 0

0 1 1 0 1 0

0 0 1 1 0 1

1 1 0 0 1 0

1 0 0 1 0 1

1 1 0 0 0 1

0 0 1 1 1 0

1 0 1 1 0 0

1 0 0 0 1 1

0 1 1 1 0 0

0 1 1 0 0 1

0 0 0 1 1 1

1 1 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 1 0 1 1 0

1 0 0 1 0 1

0 1 1 0 0 1

1 0 1 0 1 0

1 1 0 1 0 0

0 0 1 0 1 1

0 0 0 1 1 1

1 1 1 0 0 0

0 0 1 0 1 1

0 1 0 1 0 1

1 0 1 0 1 0

1 1 0 1 0 0

1 1 0 1 0 0

1 0 0 0 1 1

1 0 1 1 0 0

0 1 0 0 1 1

0 1 1 0 0 1

0 0 1 1 1 0

0 0 1 1 1 0

0 0 0 1 1 1

0 1 1 1 0 0

1 0 1 0 0 1

1 1 0 0 1 0

1 1 0 0 0 1

0 1 1 0 1 0

0 1 0 0 1 1

1 0 1 0 0 1

1 0 0 1 1 0

0 1 1 1 0 0

1 0 0 1 0 1

1 0 1 0 0 1

0 1 0 1 0 1

1 1 0 0 1 0

0 0 1 1 0 1

0 1 1 0 1 0

1 0 0 1 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

The computation of automorphism groups can be done in GAP [10] using share package GRAPE [36] with nauty [31]. The adjacency matrix of R1 is listed in Table 2 for one of these four schemes. Note that we have reordered rows and columns so that the imprimitive structure is clear. The matrix is not in maximal form in this ordering (even with the 3’s and 2’s that have been replaced by 0’s).

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A Bush-type Hadamard matrix of order 36 was ﬁrst constructed by Janko [24]. But a “skew” Bush-type Hadamard matrix was not previously known. Bussemaker, Haemers and Spence [4] proved that a symmetric Bush-type Hadamard matrix of order 36 does not exist.

5 Concluding Remarks We have seen in Sect. 3 that very few primitive non-symmetric 3-class association schemes are known. In fact (except for the ﬁrst 9 cases) the problem of existence is still open for the majority of feasible parameter sets. We do not expect that the orderly generation algorithm described in Sect. 3 can be applied to the remaining open cases in the primitive case. However, the other technique using information about the strongly regular graph obtained by merging the non-symmetric relations may still be used in some particular cases. It would also be very useful to develop new computer aided search methods or even some computer free methods. It could also be interesting to get information about existence of association schemes with a given group of automorphisms. In the imprimitive case the situation is quite diﬀerent. Here we have many constructions, especially because of the connection to Hadamard matrices. The most interesting open problem in the imprimitive case is whether there exist association schemes of type 2 with 4 ≤ r < m, other than (r, m) = (4, 16). The smallest feasible case is r = 4, m = 10 with order 40. We tried to attack this problem with the orderly generation algorithm, but it seems that the search space is too large. However, it may be that the algorithm can be improved so that this problem can be solved. But it seems that it is easier to solve the still open problem of complete enumeration of association schemes in the case m = r = 6.

Acknowledgments The author wishes to thank Prof. Mikhail Klin for some very helpful discussions and suggestions on this paper and the research reported in it. The author is pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria for their attention to this project.

References 1. E. Bannai and T. Ito, Algebraic Combinatorics. I, Benjamin-Cumming, Menlo Park, 1984.

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2. E. Bannai and S.-Y. Song, Character tables of ﬁssion schemes and fusion schemes, Europ. J. Combin., 14 (1993), 385–396. 3. A. E. Brouwer, Strongly regular graphs, in C. J. Colbourn and J. H. Dinitz (eds.) The CRC Handbook of Combinatorial Designs, pp. 667–685, CRC Press, Boca Raton, 1996. 4. F. C. Bussemaker, W. Haemers, and E. Spence, The search for pseudo orthogonal Latin squares of order six, Designs Codes Cryptogr., 21 (2000), 77–82. 5. K. Coolsaet, J. Degraer, and E. Spence, The strongly regular (45, 12, 3, 3) graphs, Electron. J. Combin. (1), 13 (2006), Research Paper 32, 9 pp. 6. E. R. van Dam, Three-class association schemes, J. Algebraic Combin., 10 (1999), 69–107. 7. H. Enomoto and R. A. Mena, Distance-regular digraphs of girth 4, J. Combin. Theory, Ser. B, 43 (1987), 293–302. 8. I. A. Faradˇzev, Constructive enumeration of combinatorial objects, in Probl`emes combinatoires et th´eorie des graphes. Colloq. Internat. CNRS, Vol. 260, pp. 131–135, CNRS, Paris, 1978. 9. I. A. Faradˇzev, M. H. Klin, and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, in I. A. Faradˇzev, A. A. Ivanov, M. H. Klin, and A. J. Woldar (eds.) Investigations in Algebraic Theory of Combinatorial Objects, pp. 1–152, Kluwer Academic, Dordrecht, 1994. 10. The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.9 ; 2006, http://www.gap-system.org. 11. R. W. Goldbach and H. L. Claasen, A primitive non-symmetric 3-class association scheme on 36 elements with p111 = 0 exists and is unique, Europ. J. Combin., 15 (1994), 519–524. 12. R. W. Goldbach and H. L. Claasen, On splitting the Clebsch graph, Indag. Math. (N.S.), 5 (1994), 285–290. 13. R. W. Goldbach and H. L. Claasen, Feasibility conditions for nonsymmetric 3-class association schemes, Discrete Math., 159 (1996), 111– 118. 14. R. W. Goldbach and H. L. Claasen, The structure of imprimitive nonsymmetric 3-class association schemes, Europ. J. Combin., 17 (1996), 23– 37. 15. D. G. Higman, Coherent conﬁgurations, Geom. Dedicata, 4 (1975), 1–32. 16. A. J. Hoﬀman and R. R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Develop., 4 (1960), 497–504. 17. Y. J. Ionin and H. Kharaghani, Doubly regular digraphs and symmetric designs, J. Combin. Theory, Ser. A, 101 (2003), 35–48. 18. N. Ito, Doubly regular asymmetric digraphs, Discrete Math., 72 (1988), 181–185. 19. N. Ito, Automorphism groups of DRADs, in Group Theory, Singapore, 1987, pp. 151–170, de Gruyter, Berlin, 1989.

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20. N. Ito, Doubly regular asymmetric digraphs with rank 5 automorphism groups, in Groups–Korea 1988. Lecture Notes in Math., Vol. 1398, pp. 94–99, Springer, Berlin, 1989. 21. N. Ito, On spectra of doubly regular asymmetric digraphs of RH-type, Graphs Combin., 5 (1989), 229–234. 22. A. A. Ivanov, M. H. Klin, and I. A. Faradˇzev, The Primitive Representations of the Non-abelian Simple Groups of Order less than 106 . Part 2 (Russian), Preprint, Moscow, Institute for System Studies, 1984, 76 pp. 23. S. Iwasaki, A characterization of PSU(3, 32 ) as a permutation group of rank 4, Hokkaido Math. J., 2 (1973), 231–235. 24. Z. Janko, The existence of a Bush-type Hadamard matrix of order 36 and two new inﬁnite classes of symmetric designs, J. Combin. Theory, Ser. A, 95 (2001), 360–364. 25. L. K. Jørgensen, Normally regular digraphs, preprint R-94-2023, Institute for Electronic Systems, Aalborg University, 1994, 44 pp. 26. L. K. Jørgensen and M. H. Klin, Switching of edges in strongly regular graphs. I. A family of partial diﬀerence sets on 100 vertices, Electron. J. Combin., 10 (2003), Research Paper 17, 31 pp. 27. L. K. Jørgensen, G. A. Jones, M. H. Klin, and S. Y. Song, Normally Regular Digraphs, Association Schemes and Related Combinatorial Structures (in preparation). 28. L. K. Jørgensen, Schur Rings and Non-symmetric Association Schemes on 64 Vertices, Preprint R-2008-17, Department of Mathematical Sciences, Aalborg University, 2008, 16 pp. 29. H. Kharaghani, On the twin designs with the Ionin-type parameters, Electron. J. Combin., 7 (2000), Research Paper 1, 11 pp. 30. R. A. Liebler and R. A. Mena, Certain Distance regular digraphs and related rings of characteristic 4, J. Combin. Theory, Ser. B, 47 (1988), 111– 123. 31. B. D. McKay, Nauty User’s Guide (Version 1.5), Technical report TRCS-90-02, Australian National University, Computer Science Department, 1990. 32. B. D. McKay and E. Spence, Classiﬁcation of regular two-graphs on 36 and 38 vertices, Australas. J. Combin., 24 (2001), 293–300. 33. A. Neumaier, New inequalities for the parameters of an association scheme, in Combinatorics and Graph Theory, Calcutta, 1980. Lecture Notes in Math., Vol. 885, pp. 365–367, Springer, Berlin, 1981. 34. R. C. Read, Every one a winner or How to avoid isomorphism search when cataloguing combinatorial conﬁgurations, Ann. Discrete Math., 2 (1978), 107–120. 35. K. B. Reid and E. Brown, Doubly regular tournaments are equivalent to skew Hadamard matrices, J. Combin. Theory, Ser. A, 12 (1972), 332–338. 36. L. H. Soicher, GRAPE: a system for computing with graphs and groups, in L. Finkelstein and W. M. Kantor (eds.) Groups and Computation, DIMACS

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Series in Discrete Mathematics and Theoretical Computer Science, Vol. 11, pp. 287–291, AMS, Providence, 1993. 37. S.-Y. Song, Class 3 association schemes whose symmetrization have two classes, J. Combin. Theory, Ser. A, 70 (1995), 1–29. 38. H. A. Wilbrink and A. E. Brouwer, A (57, 14, 1) strongly regular graph does not exist, Indag. Math., 45 (1983), 117–121.

Sets of Type (d1 , d2 ) in Projective Hjelmslev Planes over Galois Rings Axel Kohnert Mathematisches Institut, University of Bayreuth, 95440 Bayreuth, Germany. [email protected]

Summary. In this paper we construct sets of type (d1 , d2 ) in the projective Hjelmslev plane. For computational purposes we restrict ourself to planes over Zps with p a prime and s > 1, but the method is described over general Galois rings. The existence of sets of type (d1 , d2 ) is equivalent to the existence of a solution of a Diophantine system of linear equations. To construct these sets we prescribe automorphisms, which allows to reduce the Diophantine system to a feasible size. At least two of the newly constructed sets are ‘good’ u-arcs. The size of one of them is close to the known upper bound.

Key words: Projective Hjelmslev plane, Two-weight codes, Arcs

1 Introduction and Motivation The projective Hjelmslev plane over a Galois ring is a generalization of the projective plane over a ﬁnite ﬁeld GF (q) with ﬁeld size q a power of a prime p. Similar to the ﬁnite ﬁeld case the Galois ring GR(ps , psm ) is deﬁned for positive integers s, m as the ring Zps [x]/(h) where h is a monic polynomial of degree m over Zps which is irreducible over Zp . For diﬀerent choices of the polynomial h, the resulting Galois rings are isomorphic. Two limiting special cases of Galois rings are the ﬁnite ﬁelds GF (q) which are isomorphic to GR(p, q) and the modular integers Zps which are isomorphic to GR(ps , ps ). Basic facts about Galois rings can be found in [21]. For computational purposes we will restrict us to Zps in this paper. To construct the projective Hjelmslev plane we deﬁne the points as the free rank 1 submodules of GR(ps , psm )3 . The lines are the free submodules of rank 2. The incidence is given by set inclusion. In general this construction works for the larger class of chain rings R, the corresponding projective Hjelmslev plane is denoted by PHG(2, R). In this paper the ring R is always a Galois ring. Much more on projective Hjelmslev planes can be found in the work of Honold, Landjev and their coworkers [12, 13, 17, 16]. A useful tool M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 9,

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is the homomorphism φ : Zps+1 → Zps which maps an representing element from Zps+1 to its remainder modulo ps . φ can be extended to a mapping φˆ : PHG(2, GR(ps+1 , p(s+1)m )) → PHG(2, GR(ps , psm )). This function maps points to points and lines to lines. It allows to deﬁne a neighborhood of a point (or a line) in PHG(2, GR(ps+1 , p(s+1)m )) as the set of points (or lines) ˆ having the same image under φ. For two nonnegative integers d1 and d2 a set C of type (d1 , d2 ) (also called two-intersection set) in a projective Hjelmslev plane is a set of points such that every line of the plane contains either d1 or d2 points of C. We always assume d1 < d2 . In the case of a ﬁnite ﬁeld the problem of sets of type (d1 , d2 ) has been studied in a large number of papers (e.g. [9–11, 18–20]). They also study the more general case of point sets in P G(k, q) with two intersection numbers with respect to the hyperplanes. The interest in such point sets in the projective plane comes from the fact that they include hyperovals, some maximal arcs, unitals and Baer subplanes [11]. In the general case of the projective space P G(k, q) with k > 2 the sets of type (d1 , d2 ) have been also studied in the equivalent language of linear codes. Then these point sets are two-weight codes. For a survey see [8]. In coding theory one is interested in a high minimum distance for a ﬁxed length n of the code, this corresponds to a point set with n points and intersection numbers as low as possible. There are cases where the best (for coding theory) point sets are such of type (d1 , d2 ). More on the connection between linear codes and projective geometry can be found in [1, 3]. Also in the case of a projective Hjelmslev plane over a Galois ring there are links to coding theory. There are famous codes like the Nordstrom-RobinsonCode which are ‘better’ than the linear codes which are connected to P G(k, q). These better codes are Z4 -linear codes. To describe these Z4 -linear codes using projective geometry we need the projective Hjelmslev geometry. Now the hope is to ﬁnd more good codes studying PHG(k, GR(ps , psm )) in general.

2 Parameters There are several relations connecting the two parameters of the set C of type (d1 , d2 ) to the number of lines and points in PHG(2, GR(ps , psm )). These will allow to restrict the search to the cases of feasible pairs of parameters. We denote by t1 and t2 the number of lines intersecting with d1 points respectively d2 points from the set C. For a projective Hjelmslev plane over GR(ps , psm ) with point set P and line set L we know with q := pm : Lemma 1. ([16] Fact 1.) 1. |L| = |P | = (q 2 + q + 1)q 2(s−1) 2. Each line (point) is incident with (q + 1)q s−1 points (lines).

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The ﬁrst equations show that the numbers of lines and points in a Hjelmslev plane over a Galois ring is a multiple of the number of points and lines in P G(2, q). It is possible to get the Hjelmslev plane by substituting one point of P G(2, q) by q 2(s−1) points building a neighborhood in PHG(2, GR(ps , psm )). Using the lemma above we can derive the following relations with c = |C| : 1. t1 + t2 = (q 2 + q + 1)q 2(s−1) 2. d1 t1 + d2 t2 = c(q + 1)q s−1 These two equations give restrictions on possible values of d1 and d2 as t1 and t2 have to be integral numbers. In the case s = 1 (i.e. ﬁnite ﬁeld) there is a third relation, which we get by counting the number of pairs of diﬀerent points in C in two ways: 3. d1 (d1 − 1)t1 + d2 (d2 − 1)t2 = c(c − 1) The right hand is the number of diﬀerent pairs in C. The left hand side we get when we look at the unique line corresponding to the pair of points. In t1 cases this is a line having intersection number d1 . Counting the possible pairs in C corresponding to this line we get the ﬁrst summand. This last equation cannot easily be generalized to an s greater than 1 as the number of lines through a pair of diﬀerent points from C depends then on the neighbor relation between the two points. There may be more than one line through two points, which changes relation 3 into an inequality. In general it is possible to construct new sets of type (d1 , d2 ) over PHG(2, GR(ps+1 , p(s+1)m )) using two-intersection sets in PHG(2, GR(ps , psm )). A useful starting point for these recursive constructions are the single points and complete lines in P G(2, p) which is isomorphic to PHG(2, GR(p, p)) or a single point in an arbitrary projective Hjelmslev plane. Lemma 2 (Recursive construction). Let S be a set of type (d1 , d2 ) over GR(ps , psm ), then there is a set of type (pd1 , pd2 ) over GR(ps+1 , p(s+1)m ). ˆ PHG(2, GR(ps+1 , p(s+1)m )) → Proof. The key is the function φ: s sm PHG(2, GR(p , p )). It maps two-intersection sets to two-intersection sets. Each element in S is replaced by the p2 elements of the complete neighborˆ in PHG(2, GR(ps+1 , p(s+1)m )). The t1 lines hood (the preimages under φ) intersecting in d1 points are mapped to p2 t1 lines intersecting in pd1 points, and the t2 lines intersecting in d2 points are mapped to p2 t2 lines intersecting in pd2 points. Example 1. Take a line in the Fano plane P G(2, 2) = PHG(2, GR(2, 2)). This is a set of type (1, 3) with t1 = 6 and t2 = 1 and order 3. From this we construct a set of type (2, 6) in PHG(2, GR(4, 4)) with t1 = 24 and t2 = 4 and order 12.

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3 Constrution of Sets of Type (d1 , d2 ) The set P of points and the set L of lines of a projective Hjelmslev plane PHG(2, GR(ps , psm )) deﬁne an incidence system. Denote by M the corresponding incidence matrix. The rows are labeled by the lines, the columns are labeled by the points, then we have for a point p and a line l: 1 if p is incident with l, Ml,p := 0 otherwise. It is then possible to state the existence of a (d1 , d2 ) using a Diophantine system of equations: Theorem 1. There is a set of type (d1 , d2 ) in PHG(2, GR(ps , psm )) if and only if there is a 0/1-solution x = (x1 , . . . , x|P | )T of the following system of equations ⎛ ⎞ d1 or d2 ⎜ ⎟ ⎜ ⎟ .. Mx = ⎜ ⎟. . ⎝ ⎠ d1 or d2 This set of equation has to be read as follows: A solution x has the property that the product of a single with x is d1 or d2 . As we want to solve this Diophantine system using some standard method we restate it as a linear equation as follows. Denote by D the matrix of the same size as M with (d1 − d2 ) on the diagonal and 0 elsewhere. We denote by (M |D) the block matrix built from the incidence matrix M and the matrix D : ⎛ m1,1 m1,2 ⎞ m1,|P | d1 − d2 0 ... 0 0 ⎜ ⎜ ⎜ ⎜ (M |D) := ⎜ ⎜ ⎜ ⎝

m2,1

m2,2

m2,|P |

0

d1 − d2

...

0

0

. . .

. . .

. . .

. . .

. . .

..

. . .

. . .

0

0

...

d1 − d2

0

0

0

...

0

d1 − d2

..

. ..

m|L|,1

m|L|,2

. m|L|,|P |

.

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

Corollary 1. There is a set of type (d1 , d2 ) in PHG(2, GR(ps , psm )) if and only if there is a 0/1-solution x|y = (x1 , . . . , x|P | , y1 , . . . , y|L| )T of the following system of equalities ⎛ ⎞ d1 ⎜ ⎟ ⎜ ⎟ (M |D)(x|y) = ⎜ ... ⎟ . ⎝ ⎠ d1 Given the solution it is possible to read oﬀ if a line l intersects with d1 points. This is the case if yl = 0, or with d2 points, in this case yl = 1.

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The size of this problem is given by the number of points. In general this number is growing too fast. To handle also larger cases we apply the following method. We no longer look for an arbitrary set of type (d1 , d2 ). We are now only interested in a set which has a prescribed group G < PGL(2, GR(ps , psm )) of automorphisms. An automorphism ϕ of a point set C = {c1 , . . . , cn } is an element from PGL(2, GR(ps , psm ) such that C = {ϕ(c1 ), . . . , ϕ(cn )}. The main advantage of this method is that the size of the system of equations is much smaller, it will only have the size equal to the number of orbits of G on the points of PHG(2, GR(ps , psm )). We can summarize this construction as a two-step process: • In a ﬁrst step the solution of a construction problem is described as a solution of a Diophantine system of linear equations. • In a second step the size of the linear system is reduced by prescribing automorphisms. This construction method is a general approach that works for many discrete structures as designs [2, 15], q-analogs of designs [6], arcs in projective geometries [7] or linear codes [1, 4, 5]. The general method is as follows: The matrix M is reduced by adding up columns (labeled by the points of PHG(2, GR(ps , psm )) corresponding to the orbits of G. Now because of the relation p∈l ⇐⇒ ϕ(p) ∈ ϕ(l) (1) for any point p and line l and any automorphism ϕ ∈ G the rows corresponding to lines from a orbit of G are equal, therefore these are removed from the system of equations and we get a square matrix denoted by M G . The number of orbits on the points and the number of orbits on the lines is equal, as we can label the lines by the orthogonal point and then act with the transposed matrix. We denote by ω1 , . . . , ωs the orbits on the points and by Ω1 , . . . , Ωs the orbits on the lines. For an entry of M G we have: G = |{p ∈ ωj : p ∈ l}| MΩ i ,ωj

where l is a representative of the line orbit Ωi . Because of property (1) this deﬁnition is independent of the representative. Now we can restate the above theorem in a version with the condensed matrix M G : Theorem 2. Let G be a subgroup of PGL(2, GR(ps , psm )). There is a set of type (d1 , d2 ) in PHG(2, GR(ps , psm )) whose group of automorphisms contains G as a subgroup if, and only if, there is a 0/1-solution x = (x1 , . . . , x|s| )T of the following system of equations: ⎛ ⎞ d1 or d2 ⎜ ⎟ ⎜ ⎟ .. M Gx = ⎜ ⎟. . ⎝ ⎠ d1 or d2

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To solve this using a computer we transform it like in the above corollary into a Diophantine system of linear equations and using the slack variables we get the information which lines intersect in d1 points and which one in d2 points.

4 Example We describe the construction of the set of type (2, 5) over Z9 with 39 points, which is a very good 5-arc as explained in the following section with results. PHG(2, Z9 ) has 117 points, therefore the Diophantine system of equations which is to be solved would have 234 variables and 117 equations. We prescribe a group G of automorphisms generated by a single element: ⎛ ⎞ 7 1 0

⎜ ⎟ ⎟ G := ⎜ ⎝4 8 4⎠ . 5 3 8 This group has 9 orbits, each of size 13. In fact this group is a lifted version (i.e. a preimage under φ) of the Singer cycle in PGL(2, 3). PHG(2, Z9 ) can be constructed from P G(2, 3) by substituting each point in P G(2, 3) by 9 ‘lifted’ points of PHG(2, Z9 ). Each orbit now contains for each point of P G(2, 3) one lifted point. The condensed matrix M G is a 9 × 9 matrix: ⎞ ⎛ 0 3 2 2 1 1 0 1 2 ⎟ ⎜ ⎜0 0 2 2 1 1 3 1 2⎟ ⎟ ⎜ ⎟ ⎜ ⎜1 1 0 3 2 2 1 2 0⎟ ⎟ ⎜ ⎟ ⎜ ⎜2 2 1 1 0 0 2 3 1⎟ ⎟ ⎜ ⎟ ⎜ ⎟. 1 1 0 0 2 2 1 2 3 MG = ⎜ ⎟ ⎜ ⎟ ⎜ ⎜3 0 2 2 1 1 0 1 2⎟ ⎟ ⎜ ⎟ ⎜ ⎜1 1 3 0 2 2 1 2 0⎟ ⎟ ⎜ ⎟ ⎜ ⎜2 2 1 1 3 0 2 0 1⎟ ⎠ ⎝ 2 2 1 1 0 3 2 0 1 The solution x = (1, 0, 0, 0, 1, 1, 0, 0, 0) of the equation from Theorem 2 corresponds to the set of type (2, 5) with 39 points built from three orbits. From

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⎛ ⎞ 2 ⎜ ⎟ ⎜2⎟ ⎜ ⎟ ⎜ ⎟ ⎜5⎟ ⎜ ⎟ ⎜ ⎟ ⎜2⎟ ⎜ ⎟ ⎜ ⎟ G T ⎟ M x =⎜ ⎜5⎟ ⎜ ⎟ ⎜5⎟ ⎜ ⎟ ⎜ ⎟ ⎜5⎟ ⎜ ⎟ ⎜ ⎟ ⎜5⎟ ⎝ ⎠ 5 we read oﬀ which line orbits have intersection size 2 and which one size 5.

5 Results In this section we give results for projective Hjelmslev planes over the Galois rings isomorphic to Z4 , Z8 , Z9 , Z16 , Z25 , Z27 . As the complement of a set of type (d1 , d2 ) is again a set with only two intersection numbers, we list only those sets C where |C| is at most half of the points. In Table 1 we list the parameters (d1 , d2 ) of two-intersection sets we constructed with the method described. By t1 and t2 we denote the number of lines having intersection numbers d1 and d2 . We denote by ∗ in the second column if this set can not be constructed using the recursive method from 2. We do not list the trivial set consisting of one point. This list is not complete, as we only construct a two-intersection set C if we ﬁrst choose a group G such that there is a C with this group of automorphism, and secondly the resulting Diophantine system is small enough to be solved. So it may happen that further parameters (d1 , d2 ) are possible and for pairs (d1 , d2 ) already in the list there may be other sets, with diﬀerent groups of automorphisms. These results are also interesting if you look for arcs. There are at least two cases where we found improvements against previously known values for the maximal size of u-arcs. More on arcs in projective Hjelmslev planes can be found in [12]. The construction of u-arcs over Galois rings will also be covered in a forthcoming paper with M. Kiermaier. Some ﬁrst results can be found in the proceedings of the 2007 conference on optimal codes [14]. The most interesting set is the 39-set of type (2, 5) in Z9 . This is a 5-arc just one point below the upper-bound of 40 points. It improves the previously known 5-arc with 31 points. The other improvement is the 310-set of type (9, 14) in Z25 . The paper by Landjev and Honold only cover the cases with s = 2. We didn’t ﬁnd tables for Z8 and Z16 .

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Axel Kohnert Table 1. Sets of type (d1 , d2 )

R Z4

|C| 4 6∗ 7∗ 12 14∗

d1 0 0 0 2 2

d2 2 2 2 6 4

t1 16 10 7 24 14

t2 12 18 21 4 14

Z8

4

6∗ 8∗ 16 24 28∗ 28 32∗ 36∗ 36∗ 44∗ 48∗ 48∗ 48 52∗ 52∗ 56

0 0 0 0 0 2 0 2 3 2 2 2 4 4 3 4 4

2

88 76 64 64 40 84 28 72 88 60 36 24 80 96 40 68 56

24 36 48 48 72 28 84 40 24 52 76 88 32 16 72 44 56

9 30∗ 36 39∗ 39∗ 42∗

0 2 3 3 2 3

3

81 75 108 78 39 66

36 42 9 39 78 51

Z9

2 2 4 4 6 4 6 7 6 6 6 8 12 7 8 8 5 12 6 5 6

R Z16

|C| 4 6∗ 8∗ 12∗ 16 24 28∗ 32 40∗ 64 96 112 112 128 144 144 176 192 192 192 208 208 224

d1 0 0 0 0 0 0 0 0 0 0 0 4 0 4 4 6 4 8 8 4 6 8 10

d2 2 2 2 2 4 4 4 4 4 8 8 12 8 12 12 14 12 24 16 12 14 16 14

t1 400 376 352 304 352 304 280 256 208 256 160 336 112 288 240 352 144 384 320 96 160 272 224

t2 48 72 96 144 96 144 168 192 240 192 288 112 336 160 208 96 304 64 128 352 288 176 224

Z25

25 155∗ 310∗ 310∗

0

5 10 14 15

625 620 310 465

150 155 465 310

5 9 10

Acknowledgements The author is pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria for their interest to this project.

References 1. A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert, and A. Wassermann, Error-Correcting Linear Codes. Classiﬁcation by Isometry and Applications. With CD-ROM, Algorithms and Computation in Mathematics, Vol. 18, Springer, Berlin, 2006, xxix, 798 pp.

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2. A. Betten, A. Kerber, A. Kohnert, R. Laue, and A. Wassermann, The discovery of simple 7-designs with automorphism group PΓ L(2, 32), in G. Cohen et al. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC-11, Paris, France, July 17– 22, Lect. Notes Comput. Sci., Vol. 948, pp. 131–145, Springer, Berlin, 1995. 3. J. Bierbrauer, Introduction to Coding Theory, Discrete Mathematics and Its Applications, CRC Press, Boca Raton, 2005, xxiii, 390 pp. 4. M. Braun, Construction of linear codes with large minimum distance, IEEE Trans. Inform. Theory (8), 50 (2004), 1687–1691. 5. M. Braun, A. Kohnert, and A. Wassermann, Optimal linear codes from matrix groups, IEEE Trans. Inform. Theory, 12 (2005), 4247–4251. 6. M. Braun, Some new designs over ﬁnite ﬁelds, Bayreuther Math. Schr., 74 (2005), 58–68. 7. M. Braun, A. Kohnert, and A. Wassermann, Construction of (n, r)-arcs in P G(2, q), Innov. Incidence Geom., 1 (2005), 133–141. 8. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97–122. 9. M. de Finis, On k-sets of type (m, n) in projective planes of square order. Finite geometries and designs, Lond. Math. Soc. Lect. Note Ser., 49 (1981), 98–103. Proc. 2nd Isle of Thorns Conf. 1980. 10. M. de Finis, On k-sets in PG(3, q) of type (m, n) with respect to planes, Ars Comb., 21 (1986), 119–136. 11. J. W. P. Hirschfeld, Projective Geometries over Finite Fields, 2nd ed., Oxford Mathematical Monographs, Clarendon, Oxford, 1998, xiv, 555 pp. 12. T. Honold and I. Landjev, On arcs in projective Hjelmslev planes, Discrete Math. (1–3), 231 (2001), 265–278. 13. T. Honold and I. Landjev, On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic, Finite Fields Appl. (2), 11 (2005), 292–304. 14. M. Kiermaier and A. Kohnert, New arcs in projective Hjelmslev planes over Galois rings, in Fifth International Workshop on Optimal Codes and Related Topics, Balchik, pp. 112–119, 2007. 15. E. S. Kramer and D. M. Mesner, t-Designs on hypergraphs, Discrete Math., 15 (1976), 263–296. 16. I. Landjev, On blocking sets in projective Hjelmslev planes, Adv. Math. Commun. (1), 1 (2007), 65–81. 17. I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl. (1), 11 (2001), 53–70. 18. T. Penttila and G. F. Royle, Sets of type (m, n) in the aﬃne and projective planes of order nine, Designs Codes Cryptogr. (3), 6 (1995), 229–245.

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19. G. Tallini, Some new results on sets of type (m, n) in projective planes, J. Geom., 29 (1987), 191–199. 20. M. Tallini Scafati, The k-sets of type (m, n) in a Galois space Sr,q (r ≥ 2), in Colloq. Int. Teorie Comb., Roma 1973, Tomo II, pp. 459–463, 1976. 21. Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientiﬁc, River Edge, 2003, x, 342 pp.

A Construction of Designs from PSL(2, q) and PGL(2, q), q = 1 mod 6, on q + 2 Points Izumi Miyamoto Department of Computer Science and Media Engineering, University of Yamanashi, Kofu 400-8511, Japan. [email protected]

Summary. Let G = PSL(2, q) or PGL(2, q). We consider the action of G on the projective line together with one additional point, which is ﬁxed by G. Assume q ≡ 1 1 (q −1)(q −3)(q −5). We construct 3-(q +2, 12 (q −1), λq ) designs mod 6 and set λq = 24 admitting PSL(2, q) as their automorphisms, if q ≡ 3 mod 4. We also construct 3(q + 2, 12 (q − 1), 2λq ) designs admitting PGL(2, q) as their automorphisms. These designs may not be simple.

Key words: Block design, Superscheme, Permutation group, Homogeneous group

1 Introduction For q = 3 mod 4, PSL(2, q) is 3-homogeneous on the q + 1 points of the projective line. So a union of certain orbits of PSL(2, q) acting on the k-subsets of the projective line forms a 3-design. Most of such designs are classiﬁed in [1]. In the present paper we consider the action of PSL(2, q) on the projective line together with one additional point, which is ﬁxed by PSL(2, q). We will construct 3-designs from unions of certain orbits of PSL(2, q) acting on 1 2 (q − 1) points, when q = 1 mod 3. The author considered a combinatorial approach to doubly transitive permutation groups as transitive extensions using some superschemes [8–10]. Superschemes are introduced in [6, 11]. Although PSL(2, q) does not extend to a triply transitive group on q + 2 points, we obtained some superschemes on q + 2 points which have properties similar to those of triply transitive groups. Such a superscheme gives a certain number of subsets Xj of distinct 4-tuples of q + 2 points. The subsets form a partition of the distinct 4-tuples. In a superscheme we consider projections π1 (x, y, z, w) = (y, z, w), similarly, π2 , . . . , π4 . In the above superschemes the projection πi , i = 1, 2, 3, 4, maps every subset Xj to the entirety of distinct triples, and |πi−1 (x, y, z)| is constant for any distinct triple (x, y, z). So we can consider an orbit-like set {w|(x, y, z, w) ∈ Xj } for any ﬁxed three distinct M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 10,

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points x, y and z. The present research is motivated by the expectation that an orbit-like set may form a 3-design, since PSL(2, q) is 3-homogeneous if q = 3 mod 4. As a result we can construct 3-designs even if there do not exist superschemes like those arising from triply transitive groups. The subgroups of PSL(2, q) are well known. Readers may refer to [3, 5]. The subgroups of G = PSL(2, q) or PGL(2, q) may also be found in [1, 2], together with results of 3-designs constructed from certain orbits of G acting on k-subsets. Data of the ﬁxed points of subgroups are also listed in [1, 2]. Some data for t-designs, t ≥ 3, appear in [7].

2 Deﬁnitions and Notations Let t, v, k and λ be positive integers satisfying that t ≤ k ≤ v and λ > 0. the set of all k-subsets of X. Let Let X be a set of points, and denote by X k X B be a speciﬁed collection from k . A k-subset in B will be called a block. We allow the possibility that B be a multi-set, i.e., B may contain repeated blocks. Then a pair (X, B) is called a t-(v, k, λ) design if every t-subset of X is contained in exactly λ blocks. If B contains no repeated blocks, then the design is called simple. Note that we may describe a design simply by indicating its set B of blocks. Let q be a prime power, and let P be the union of the Galois ﬁeld GF (q) and {∞}. For a, b, c, d ∈ GF (q), a mapping g on P is deﬁned by g : x →

ax + b , cx + d

if ad−bc = 0, x = −d/c, and g(∞) = a/c, g(−d/c) = ∞, if c = 0, and g(∞) = ∞, if c = 0. Then the set of all such mappings becomes a permutation group, denoted PGL(2, q), and we refer to the set P as the projective line. We use the GAP system [4] to compute permutation groups in our experiments. In the GAP library, a permutation group always acts on the point set {1, 2, . . . , n}. Thus we shall denote the projective line P = {1, 2, . . . , q + 1} below. Let PSL(2, q) be the subgroup of PGL(2, q) consisting of those mappings which satisfy ad − bc = 1. Let G be a permutation group on a set X. For points x, y, . . . , z of X the pointwise stabilizer of x, y, . . . , z in G is the subgroup of G consisting of all the elements g such that xg = x, y g = y, . . . , z g = z. For a subset Y of X the setwise stabilizer of Y in G is the subgroup consisting of all elements g such that y g ∈ Y for all y ∈ Y . We deﬁne the induced action of G on the set of s-subsets of X by {x1 , x2 , . . . , xs }g = {xg1 , xg2 , . . . , xgs }. We call G shomogeneous if for any pair of s-subsets there exists an element g which moves one to the other. Then PGL(2, q) is 3-homogeneous on P, and PSL(2, q) is 3-homogeneous on P if q = 3 mod 4.

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3 Construction of Designs Let G = PSL(2, q) or PGL(2, q) acting on the q+1 points of the projective line P = {1, 2, . . . , q + 1}. Then G is of order 12 g0 (q + 1)q(q − 1), where g0 = 1 or 2 according to whether G = PSL(2, q) or PGL(2, q). We consider an additional point, denoted q + 2. If G = PSL(2, q), we assume that q = 3 mod 4, in which case G is 3-homogeneous. Suppose furthermore that q = 1 mod 3. Let H be a subgroup of G of order 3. Since every nonidentity element of G has at most two ﬁxed points, H has two ﬁxed points. Suppose that H ﬁxes the points 1 and 2. Let b1 be the union of some 16 (q − 1) orbits of H of size 3, and let b2 be the union of {1, 2, q + 2} and some 16 (q − 7) orbits of H of size 3. Then both b1 and b2 consist of 12 (q − 1) points. Let G1 be the setwise stabilizer in G of b1 , and let G2 be that of b2 . Let 3g1 and 3g2 be the orders of G1 and G2 , respectively. Let K be the stabilizer of the points 1 and 2 in G. Then K is cyclic of order 12 g0 (q − 1), and the subgroup of K of order 12 (q − 1) has two orbits of length 12 (q − 1). Let b3 be one of these orbits, and let G3 be the stabilizer in G of b3 . Then the order of G3 is 12 g0 (q − 1). The blocks of our design are generated from the sets b1 , b2 and b3 under the action of G. Let Bi denote the set of blocks generated from bi via the action of G on the 1 1 2 (q − 1)-subsets of P∪{q + 2}. Thus |Bi | = 6 (g0 /gi )(q + 1)q(q − 1) for i = 1, 2 1 1 and |B3 | = ( 2 g0 (q + 1)q(q − 1))/( 2 g0 (q − 1)) = (q + 1)q. For our design, the blocks generated from bi are repeated gi times. Such a set of blocks is denoted 1 (q −1)(q −3)(q −5). Then we have the following theorems. by gi Bi . Set λq = 24 Theorem 1. Suppose that G = PSL(2, q), where q = 3 mod 4 and q = 1 mod 6. Then the block set g1 B1 ∪ g2 B2 ∪ B3 forms a 3-(q + 2, 12 (q − 1),λq ) design. If g1 = g2 = 1 and q > 7, then the design is simple. Theorem 2. Suppose that G = PGL(2, q), where q = 1 mod 6. Then the block set g1 B1 ∪ g2 B2 ∪ 2B3 forms a 3-(q + 2, 12 (q − 1),2λq ) design. If g1 = g2 = 2 and q > 7, then the block set B1 ∪B2 ∪B3 gives a simple 3-design with λ = λq . Proofs. We have |g1 B1 | = |g2 B2 | = 16 g0 (q + 1)q(q − 1) blocks from each of b1 and b2 , and |g0 B3 | = g0 (q + 1)q blocks from b3 . Both b1 and b3 contain 1 48 (q − 1)(q − 3)(q − 5) distinct 3-subsets of {1, 2, . . . , q + 1}, and b2 contains 1 1 48 (q − 3)(q − 5)(q − 7) such subsets. Moreover, b2 contains 8 (q − 3)(q − 5) distinct 3-subsets of the form {x, y, q + 2}, where x, y ∈ {1, 2, . . . , q + 1}. Since G is 3-homogeneous on {1, 2, . . . , q + 1}, a counting argument reveals that every distinct 3-subset of {1, 2, . . . , q + 1} is contained in the following number of blocks:

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1 1 (|g1 B1 | + |g0 B3 |) 48 (q − 1)(q − 3)(q − 5) + |g2 B2 | 48 (q − 3)(q − 5)(q − 7) 1 (q + 1)q(q − 1) 6 1 1 ( 6 (q − 1) + 1)(q − 1)) + g0 (q + 1)q(q − 3)(q − 5)( 48 1 6 (q + 1)q(q − 1) 1 (g0 (q − 3)(q − 5) ((q − 1) + 6 + q − 7)) = 48 1 (g0 (q − 1)(q − 3)(q − 5)). = 24

=

1 6·48 (q

− 1)(q − 7))

Similarly, for x, y ∈ {1, 2, . . . , q + 1} the number of blocks containing each distinct 3-subset {x, y, q + 2} is 1 6·8 g0 (q

+ 1)q(q − 1)(q − 3)(q − 5) 1 = (g0 (q − 1)(q − 3)(q − 5)). 1 24 (q + 1)q 2

So we have obtained a 3-design. If g0 = g1 = g2 = 1, then we have a simple design. We also note that if g0 = g1 = g2 = 2 then, without repetition, we also have a simple design from the above argument.

4 Experiments We used GAP system [4] to compute our examples. Let b1 , b2 , b3 , G1 , G2 and G3 be as in the previous section. In Example 1 and 2 in Table 1, G = PSL(2, 19) = PrimitiveGroup(20,1) in the GAP library. G is generated by Table 1. Examples Example 1. G = PSL(2, 19) = PrimitiveGroup(20, 1) b1 = {4, 6, 7, 10, 12, 13, 16, 18, 19} b2 = {1, 2, 5, 8, 11, 14, 17, 20, 21} b3 = {3, 5, 7, 9, 11, 13, 15, 17, 19}

|G1 | = 3 |G2 | = 6 |G3 | = 9

Example 2. G = PSL(2, 19) = PrimitiveGroup(20, 1) b1 = {4, 5, 6, 10, 11, 12, 16, 17, 18} b2 = {1, 2, 5, 6, 11, 12, 17, 18, 21} b3 = {3, 5, 7, 9, 11, 13, 15, 17, 19}

|G1 | = 3 |G2 | = 6 |G3 | = 9

Example 3. G = PSL(2, 31) = PrimitiveGroup(32, 4) b1 = {3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 21, 26, 27, 29, 30} b2 = {1, 2, 3, 4, 5, 6, 9, 10, 11, 14, 21, 24, 26, 32, 33} b3 = {3, 6, 9, 10, 11, 13, 14, 20, 21, 23, 24, 25, 28, 31, 32}

|G1 | = 3 |G2 | = 3 |G3 | = 15

Example 4. G = PGL(2, 25) = PrimitiveGroup(26, 2) b1 = {3, 4, 7, 8, 9, 13, 14, 16, 17, 21, 22, 23} b2 = {1, 2, 3, 4, 6, 7, 9, 11, 20, 21, 23, 27} b3 = {3, 4, 5, 7, 9, 10, 15, 18, 21, 23, 24, 26}

|G1 | = 6 |G2 | = 6 |G3 | = 24

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(3, 19, 17, 15, 13, 11, 9, 7, 5)(4, 20, 18, 16, 14, 12, 10, 8, 6) and (1, 2, 12)(3, 11, 13) (4, 17, 6)(5, 14, 8)(7, 20, 18)(10, 19, 16). We construct two 3-designs. The block b3 is common in both of the two 3-designs. The blocks in B2 are duplicated. Then we have two 3-(21, 9, 168) designs. In this case, gathering all the blocks of these two designs (including repetition) gives the block set of a 4-design, namely a 4-(21, 9, 112) design. In Example 3, G = PSL(2, 31) = PrimitiveGroup(32,4). G is generated by (1, 2, . . . , 31) and (1, 32)(2, 31)(3, 16)(4, 11)(5, 24)(6, 7)(8, 23)(9, 28)(10, 25) (12, 15)(13, 19)(14, 20)(17, 30)(18, 21)(22, 29)(26, 27). In this example we get a simple 3-(33, 15, 910) design. In Example 4, G = PGL(2, 25) = PrimitiveGroup(26,2). G is generated by (3, 6, 7, 17, 10, 25, 26, 8, 23, 11, 21, 22, 5, 12, 18, 14, 9, 20, 4, 13, 24, 19, 15, 16) and (1, 19, 18, 23, 16, 17, 9, 25, 11, 3, 4, 22, 2)(5, 12, 21, 7, 10, 26, 15, 13, 24, 8, 20, 14, 6). In this example we get a simple 3-(27, 12, 440) design. Acknowledgments The author is pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria for partial support of this project.

References 1. P. J. Cameron, H. R. Maimani, G. R. Omidi, and B. Tayfeh-Rezaie, 3Designs from PSL(2, q), Discrete Math., 306 (2006), 3063–3073. 2. P. J. Cameron, G. R. Omidi, and B. Tayfeh-Rezaie, 3-Designs from PGL(2, q), Electron. J. Combin., 13 (2006), #R50. 3. L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Dover, New York, 1958. 4. The GAP Groups, Gap – Groups, Algorithms and Programming, Version 4, Lehrstuhl D f¨ ur Mathematik, Rheinisch Westf¨ alische Technische Hochschule, Aachen, Germany and School of Mathematical and Computational Sciences, Univ. St. Andrews, Scotland, 2000. 5. B. Huppert, Endliche gruppen I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer, Berlin, 1967. 6. K. W. Johnson, and J. D. H. Smith, Characters of ﬁnite quasigroups IV: products and superschemes, Europ. J. Combin., 10 (1989), 257–263. 7. G. B. Khosrovshahi and R. Laue, t-Designs with t ≥ 3, in C. J. Colbourn, and J. H. Dinitz (eds.) Handbook of Combinatorial Designs, 2nd edition, CRC Press Series on Discrete Mathematics and Its Applications, pp. 79– 101, CRC Press, Boca Raton, 2006. 8. I. Miyamoto, A generalization of association schemes and computation of doubly transitive groups, RIMS Kokyuroku on Computer Algebra –

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Izumi Miyamoto

Algorithms, Implementations and Applications, 1394 (2004), 185–189 (in Japanese). 9. I. Miyamoto, A computation of some multiply homogeneous superschemes from transitive permutation groups, in C. W. Brown (ed.) Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, pp. 293–298, ACM, New York, 2007. 10. I. Miyamoto, A combinatorial approach to doubly transitive permutation groups, Discrete Math., 308 (2008), 3073–3081. 11. J. D. H. Smith, Association schemes, superschemes, and relations invariant under permutation groups, Europ. J. Combin. (3), 15 (1994), 285–291.

Approaching Some Problems in Finite Geometry Through Algebraic Geometry G. Eric Moorhouse Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA. [email protected]

Summary. In the study of ﬁnite geometries one often requires knowledge of the ranks of related (0, 1)-incidence matrices. We describe some of the combinatorial questions in ﬁnite geometry for which formulas for these ranks are useful; and we describe methods from algebraic geometry that are useful in obtaining such rank formulas.

Key words: p-Rank, Polar space, Ovoid, Spread, Hilbert function

1 Motivation and Background Here we recall the deﬁnitions of a few standard notions from ﬁnite geometry. Considering the audience for this presentation, many of whom are coding theorists, the coding-theoretic interpretations of our objects of study, and of our results, will occasionally be explicitly mentioned. For a more extensive summary description of the relevant deﬁnitions from ﬁnite geometry, see e.g. [9, 17, 32]. We denote by P n (Fq ) the classical projective n-space over the ﬁnite ﬁeld : Fq of order q, i.e. the incidence system formed by the subspaces of Fn+1 q the points, lines, planes, etc. being the subspaces of dimension 1, 2, 3, etc.; are the projective k-subspaces. In thus the vector (k+1)-subspaces of Fn+1 q particular P 2 (Fq ) denotes the classical (i.e. Desarguesian) projective plane of order q. Non-classical projective planes exist, but all projective spaces of dimension n ≥ 3 are classical. An ovoid in projective 3-space P 3 (Fq ) is a set of q 2 + 1 points with no three collinear. Alternatively, one may consider a linear [n, 4]-code C over Fq such that the dual code C ⊥ has minimum weight ≥ 4; in this case n ≤ q 2 + 1, and equality holds iﬀ a generator matrix for C has as its columns the points of an ovoid in P 3 (Fq ). For q odd, every ovoid is an elliptic quadric (Barlotti [2]; Panella [28]). For q even, the known ovoids are the elliptic quadrics and (for q = 22e+1 ) the Suzuki-Tits ovoids. M. Klin et al. (eds.), Algorithmic Algebraic Combinatorics and Gr¨ obner Bases, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-642-01960-9 11,

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A spread in P 2n−1 (Fq ) is a set S consisting of q n +1 projective (n−1)subspaces which partition the points. These exist for all n ≥ 1 and prime powers q, and every such spread gives a plane (aﬃne or projective) of order q n known as a translation plane. This construction is responsible for most of the explicitly known ﬁnite projective planes. An orthogonal (resp., unitary) polar space is the incidence system formed by the subspaces of projective space which lie on a given nondegenerate quadric (resp., Hermitian variety). A symplectic polar space is the incidence system formed by the totally isotropic subspaces of a projective space with respect to a nondegenerate alternating form. Let P be any ﬁnite classical polar space (i.e. a ﬁnite orthogonal, unitary or symplectic polar space). An ovoid in P is a set O consisting of points of O such that every maximal subspace of P contains exactly one point of P. A spread in P is a set S consisting of maximal subspaces of P, such that every point of P lies in exactly one member of S. These notions of ovoid and spread are distinct from (albeit related to) the notions of ovoid and spread for projective spaces. In the polar space setting, one has a bipartite incidence graph formed by incidences between points and maximal subspaces. Whenever one has a bipartite graph with partition A ∪ B of the vertices (and every edge of the graph has one end in A and the other in B) then one may ask for a subset O ⊆ A such that every vertex in B is adjacent to exactly one member of O; or a subset S ⊆ B such that every vertex in A is adjacent to exactly one member of S. At this level of abstraction we see that ovoids and spreads are very similar notions. Questions of existence and possible constructions of ovoids and spreads in the ﬁnite classical polar spaces are in many cases open; for an almost-current survey see [18, pp. 345–348]. Ovoids of polar spaces are most intensively studied in the orthogonal case. If Q is a nondegenerate quadric in P 2n−1 (Fq ) then Q is hyperbolic or elliptic according as maximal subspaces lying in the quadric have projective dimension n−1 or n−2. In the hyperbolic case an ovoid in Q (deﬁned as above) is simply a set O consisting of q n−1 +1 points of Q, no two on a line of the quadric. It is known [33] that ovoids do not exist in the elliptic case for n ≥ 5; and [15] that no ovoids exist in nondegenerate quadrics of P 2n (Fq ) (the parabolic case) for n ≥ 4. There are many connections between the various notions of spreads, ovoids, and other objects. The following sample of such connections is not exhaustive but is intended to hint at the central role played by these notions in ﬁnite geometry: Spreads of projective 3-space are equivalent to ovoids in the Klein quadric (i.e. the hyperbolic quadric in projective 5-space) via the Klein correspondence (the Pl¨ ucker map). Ovoids and spreads in higher-dimensional spaces often give ovoids and spreads in lower-dimensional spaces, by a simple process of ‘slicing’. The E8 root lattice gives rise to numerous constructions (see [12, 23, 24, 26]) of ovoids in the hyperbolic quadric in projective 7-space, and these are in turn equivalent to spreads of the same quadric. Figure 1 depicts some of the connections arising between the ﬁnite geometric objects we have mentioned.

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Fig. 1. Some connections between ﬁnite geometric objects

The most signiﬁcant open question in this area is the question of whether there exist ovoids in nondegenerate quadrics in P n (Fq ) for n > 7. Ovoids in higher dimensions would give rise to signiﬁcant numbers of ovoids in dimensions 5 and 7, which seems unlikely; yet no proof of impossibility is known. One may be tempted to mimic the construction of ovoids from the E8 root lattice, using the Leech lattice to produce ovoids in quadrics in P 23 (Fp ); however this approach cannot succeed for primes p < 59 by virtue of Corollary 1 below. Such nonexistence results for ovoids motivated our interest in the p-rank formulas of Sect. 4. Another motivation for our work is the desire to better understand the following striking parallel between diﬀerent spaces admitting ovoids. 1.1 Ovoids in P 3 (Fq ), q = 2r Here the known ovoids belong to two inﬁnite families, each admitting a doubly transitive subgroup of PGL(4, q): the elliptic quadrics (for all q even, stabilized by PSL(3, q)) and the Suzuki-Tits ovoids (for q = 22e+1 only, admitting the Suzuki group 2 B2 (q)). The binary code spanned by the (characteristic vectors of the) planes of P 3 (Fq ) has dimension q 2 +1 (only for q even), and the tangent planes to any ovoid form a basis for the code. 1.2 Ovoids in Hyperbolic Quadrics in P 7 (Fq ), q = 2r Aside from one known sporadic example for q = 8 (Dye’s ovoid; see [19]) just two inﬁnite families of ovoids are known, each admitting a doubly transitive subgroup of P Ω + (8, q): one family (for all q even) stabilized by PSL(3, q), and the other (for q = 22e+1 only) admitting P SU (3, q). The binary code spanned by the (characteristic vectors of the) tangent hyperplanes to the quadric has dimension q 3 +1 (only for q even), and the tangent hyperplanes to any ovoid form a basis for the code. 1.3 Ovoids in Nondegenerate (Parabolic) Quadrics in P 6 (Fq ), q = 3r Here the known ovoids belong to two inﬁnite families, each admitting a doubly transitive subgroup of P Ω(7, q): one family (for all q = 3r ) stabilized by

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PSU (3, q), and the Ree-Tits ovoids (for q = 32e+1 only) admitting the Ree group 2 G2 (q). The ternary code spanned by the (characteristic vectors of the) tangent hyperplanes to the quadric has dimension q 3 +1 (only for q = 3r ), and the tangent hyperplanes to any ovoid form a basis for the code. Without the p-rank formulas of Sect. 4 this analogy is not complete. One may hope to use the tightness of the p-rank bound in each case to classify ovoids in each situation, or to look to the recent literature on case 1.1 (for example Brown [6, 7]) in the hopes of ﬁnding techniques that may apply also to cases 1.2 and 1.3.

2 p-Ranks Related to Projective Spaces Let A be the (0, 1)-incidence matrix of a ﬁnite point-block incidence structure, i.e. the matrix having rows indexed by points and columns indexed by blocks, and with entries 0 and 1 corresponding to nonincident and incident pointblock pairs, respectively. By the p-rank of the incidence structure, we mean the rank of A over a ﬁeld of characteristic p. It has long been known (see [13, 20, 31]) that the symmetric design of points and hyperplanes of P n (Fq ) has p-rank equal to r p+n−1 +1 n where q = pr . The binomial coeﬃcient appearing in the latter formula is in fact the coeﬃcient of tp−1 in the binomial series n+1 n+2 2 n + p − 1 p−1 1 =1+ t+ t + ··· + t + ··· 1 2 p−1 (1 − t)n+1 which arises as the Hilbert series for projective n-space. More explanation of the connection between p-ranks and Hilbert series is given in Sect. 3. Stronger information is in fact available: Black and List [3] give the Smith normal form of the point-hyperplane incidence matrix, although here we omit the details. More generally, one may ask for the p-rank the design of points versus projective (n−k)-subspaces of P n (Fq ) where q = pr ; that is, the dimension of the linear code C = C(n, k, pr ) spanned over Fp by the (characteristic vectors of the) projective subspaces of P n (Fq ) of codimension k. The ﬁrst formula available for this is that of Hamada [16]; see also [5, p. 366]. Unfortunately the computational time required to evaluate Hamada’s formula can be prohibitive, even for rather modest values of the input parameters. Fortunately however, dim C can be computed quite easily using the information implicit in [1], where the structure of the code as an Fq G-module for the group G = PGL(n+1, Fq ) is given. We have dim C(n, k, pr ) = 1 + coeﬀ. of tr in tr[(I − tM )−1 ]

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where M is the k × k matrix with (i, j)-entry equal to the coeﬃcient of tpi−j in (1 + t + t2 + · · · + tp−1 )n+1 . For example the following MapleTM code [21] determines the dimension of the F5 -code spanned by lines of P 3 (F5r ): > > > > > > > > > >

with(linalg): p:=5: n:=3: k:=2: S:=simplify(((1-t^p)/(1-t))^(n+1)): M:=array(1..k,1..k): for i from 1 to k do for j from 1 to k do M[i,j]:=coeff(S,t,p*j-i): od: od: print(M);

35 80

20 85 > simplify(trace(inverse(\&*()-t*M))); −

2(60t − 1) 1375t2 − 120t + 1

> series(\%,t=0,6);

2 + 120t + 11650t2 + 1233000t3 + 131941250t4 + 14137575000t5 + O(t6 ) Thus the dimension of the code spanned by the lines of P 3 (F5r ) is 120, 11650, 1233000, 131941250, 14137575000, . . . for r = 1, 2, 3, . . . . Again, even stronger information is available [10] from the Smith normal form of the incidence matrix of points versus projective (n−k)-subspaces.

3 p-Ranks via the Hilbert Function Consider the Fq -linear code Cˆ of length N = (q n+1 − 1)/(q − 1) spanned by the (characteristic vectors of the) hyperplanes of P n (Fq ) where q = pr . ˆ The subcode hyperplanes has of the C ⊂r C spanned by the complements ˆ itself has dimension p+n−1 r + 1. Now let V be , while C dimension p+n−1 n n a subset of the points of P n (Fq ). Denote by CˆV and CV the punctured codes

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of length |V| obtained by simply restricting Cˆ and C (respectively) to the coordinate positions indexed by V. We are interested in general methods for determining the dimensions of CˆV and CV . We note that CV ⊆ CˆV is a subcode of codimension at most 1, and in many cases of interest (see Theorem 3 below) the exact codimension is 1. So for now we focus attention on CV . We are most interested in the case of a point set V arising as the set of Fq -rational points of a projective variety. We establish notation to describe this case. Consider the polynomial ring R = Fq [X0 , X1 , . . . , Xn ] = d≥0 Rd where Rd is the d-homogeneous part of R with respect to the standard degree grading. Let I ⊆ R be a homogeneous ideal, and let V be the set of Fq -rational points of projective n-space where I vanishes; thus V = V(I +J) is the zero set of the ideal I + J where J is generated by the polynomials Xiq Xj − Xi Xjq for 0 ≤ i < j ≤ n. Let I = I(V) ⊆ R be the ideal generated by all homogeneous √ polynomials vanishing on V; this is just the radical ideal I = I + J. We denote the Hilbert function of I by hI (d) = dim(Rd /Id ) where Id = I ∩ Rd . Denote by LM (I) the set of leading monomials in I with respect to some ﬁxed monomial ordering. A monomial in R is standard if it is not in I. Then hI (d) is the number of standard monomials of degree d. For simplicity we consider ﬁrst the special case q = p. Theorem 1. (See [27].) If q = p then dim(CV ) = hI (p − 1). 3.1 Computational Example Consider the incidence system of points of the cubic surface x3 + y 3 + z 2 w = 0 in P 3 (F13 ) versus all hyperplanes of the projective space. Using Macaulay 2 [14] we compute i1 i2 i3 i4 i5 o5 i6 o6 i7 o7 i8

: : : : : : : : : : :

o8 =

p = 13; F = ZZ/(p); R = F[x,y,z,w]; S = (s,t)->s^p*t-s*t^p; J = ideal(S(x,y),S(x,z),S(x,w),S(y,z),S(y,w),S(z,w)); Ideal of R I = ideal(x^3+y^3+z^2*w)+J; Ideal of R II = radical(I); Ideal of R hilbertSeries(I) 1 − T 3 − 6T 14 + 4T 15 − T 16 + 6T 17 − 4T 18 + T 19 + 2T 26 + · · · + 2T 33 (1 − T )4

o8 : Divide i9 : hilbertSeries(II) o9 =

1 − T 3 − 2T 10 − 4T 11 + T 12 + 9T 13 − 3T 14 + 2T 15 − T 16 − 2T 17 (1 − T )4

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o9 : Divide i10 : hilbertSeries(II, Order=>p) o10 = 1 + 4T + 10T 2 + 19T 3 + 31T 4 + 46T 5 + 64T 6 + · · · +187T 11 + 200T 12 o10 : ZZ [T, MonomialOrder => RevLex, Inverses => true]} From the coeﬃcient of T p−1 we see that dim(CV ) = 200, and so dim(CˆV ) = 201 by Theorem 3 below. The most time-consuming step in this example (the computation of the radical ideal) requires at most a few seconds on a typical personal computer, but in other examples this step may overwhelm the computational resources of the machine. In such cases one might try to explicitly determine the radical by other means; or it may be necessary to determine the required p-rank by Gaussian elimination. For example we check independently that the above cubic surface has 209 points, and that the 209× 2380 incidence matrix of points versus hyperplanes has 13-rank equal to 201. Now consider the general case q = pr , r ≥ 1. We deﬁne a p-standard 2 monomial to be a monomial of the form m0 mp1 mp2 · · · , a ﬁnite product in which each mi is a standard monomial of degree less than p. (This deﬁnition is not standard; sorry, no pun intended!) Denote by h†I (d) the number of p-standard monomials of degree d. Theorem 2. (See [27].) dim(CV ) = h†I (q − 1). This requires us to count the number of monomials of the form 2

r−1

m0 mp1 mp2 · · · mpr−1

where each mi is a standard monomial of degree p − 1. 3.2 Example: Projective n-Space Let I = 0, so that I = 0 and V consists of all (q n+1 − 1)/(q − 1) points of P n (Fq ). Every monomial is standard, and the p-standard monomials are those 2

r−1

of the form m0 mp1 mp2 · · · mpr−1 where each monomial mi has degree p − 1. There are p+n−1 choices for each mi , and hence the number of p-standard n r . This gives the well-known monomials of degree q − 1 is h†I (q − 1) = p+n−1 n value for dim(CV ); and the value dim(CˆV ) = 1 + dim(CV ) may be seen as a special case of the following (for a vacuous set of k = 0 polynomials). Theorem 3. Let f1 , . . . , fk ∈ R be nonconstant homogeneous polynomials of total degree i deg(fi ) ≤ n − 2, and let V be the set of all points in P n (Fq ) where every fi vanishes. Then dim(CˆV /CV ) = 1.

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Proof. Let M be the number of vectors in Fn+1 where all f1 , . . . , fk vanish. q Since the total degree i deg(fi ) < n, the Chevalley-Warning Theorem [30, p. 5] shows that p divides M . But the homogeneity of f1 , . . . , fk means that q − 1 divides M − 1, so in fact M = mp(q − 1) + q for some m ≥ 0. Thus |V| = (M − 1)/(q − 1) = mp + 1 ≡ 1 mod p. Now let h ∈ R1 be a nonzero homogeneous linear polynomial, and let where all k + 1 of the polynomials Mh be the number of vectors in Fn+1 q f1 , f2 , . . . , fk , h vanish. Since the total degree again satisﬁes 1 + i deg(fi ) ≤ n − 1, the previous argument also shows that Mh = mh p(q − 1) + q for some mh ≥ 0. Thus |H ∩ V| = mh p + 1 ≡ 1 mod p where H is the hyperplane of P n (Fq ) consisting of all points where h vanishes. ˆ Since dim(C/C) = 1, it suﬃces to ﬁnd a nonzero linear functional φ : CˆV → Fp vanishing on CV . For v ∈ CˆV , deﬁne φ(v) to be simply the sum of the coordinate entries of v. In case v is the characteristic vector of a hyperplane H (restricted to V), we have φ(v) = |H ∩ V| ≡ 1 mod p. Similarly if v is the characteristic vector of the complement of a hyperplane H, then φ(v) = |V| − |H ∩ V| ≡ 1 − 1 ≡ 0 mod p. Since we have considered typical generators for the codes CˆV and CV , our φ has the required properties and the conclusion follows.

4 p-Ranks Related to Polar Spaces and Grassmannians We survey some interesting p-rank formulas and some applications to bounds for ovoids. Each p-rank formula listed here is derived either by the approach described in Sect. 3, or from the theory of group representations. Our notation q, p, n, R, J, CV , etc. is the same as in Sect. 3. Theorem 4. (See [4].) Let I = (Q) where Q(X0 , X1 , . . . , Xn ) ∈ R2 is an irreducible quadratic form, and let Q = V(I + J) be the resulting quadric in P n (Fq ). Let CˆQ be the Fq -linear code of length |Q| spanned by the hyperplane sections of the quadric. Then

r p+n−1 p+n−3 ˆ + 1. dim(C(Q)) = − n n If a nondegenerate quadric Q in P n (Fq ) admits an ovoid O then the tangent hyperplanes to Q at the points of O span a subcode of CˆQ of dimension pn/2r + 1. This gives Corollary 1. (See [4].) There do not exist ovoids in Q (using the notation of Theorem 4) if p+n−1 p+n−3 n/2 > − . p n n In particular ovoids do not exist in Q for n = 9 and p = 2, 3; or for n = 11 and p = 2, 3, 5, 7.

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In studying ﬁnite projective planes, it is often useful to have an explicit basis for the Fp -linear code spanned by the lines. In the case of classical +1 and so it had (Desarguesian) planes P 2 (Fp ), this code has dimension p+1 2 long been speculated that an explicit basis could be formed from any conic (which has p + 1 points). This follows also from our approach to p-ranks: Corollary 2. (See [4].) Let Cˆ be the Fp -linear code spanned by the lines of + 1. Let C ⊂ Cˆ be the subcode of dimension P 2 (Fp ), so that dim(C) = p+1 2 p+1 spanned by the complements of the lines. Let Q be any conic in the 2 plane, so that Q has p + 1 tangent lines, p+1 secant lines, and p2 passant 2 lines (i.e. lines not meeting Q). Then the complements to the secants form a basis for C. Moreover the tangents and the passants together form a basis ˆ for C. We remark in passing that another choice of explicit basis is found in [22]. Theorem 5. (See [25].) Suppose q = q02 = pr (r even) and let I = (U ) where U ∈ Rq0 +1 is a nondegenerate unitary form; we may choose coordinates so that U (X0 , X1 , . . . , Xn ) = X0q0 +1 + X1q0 +1 + · · · + Xnq0 +1 . Let H = V(I + J) be the resulting hermitian variety in P n (Fq ). Let CˆH be the Fq -linear code of length |H| spanned by the hyperplane sections of H. Then dim(CˆH ) =

p+n−1 n

2

p+n−2 − n

2 r/2 + 1.

Bounds for ovoids in unitary polar spaces, similar to those of Theorem 4 and Corollary 1, are obtained [25] using Theorem 5. Theorem 6. Let Q be a nondegenerate (parabolic) quadric in P 4 (Fq ), and consider the incidence system of points of P 4 (Fq ) versus lines of Q. The prank of this incidence system is √ 2r 1−√17 2r + ; (a) (for q = 2r ) 1 + 1+2 17 2 (b) (for q = p) (c) (for q = pr )

1+

p(p+1)2 ; 2

r r 1 + α+ + α− where α± =

p(p+1)2 4

±

p(p2 −1) √ 17. 12

The incidence system of Theorem 6 is a classical generalized quadrangle of order (q, q); and it immediately follows that the dual generalized quadrangle also has p-rank as given by Theorem 6. This dual generalized quadrangle is the symplectic polar space in P 3 (Fq ) formed by a nondegenerate alternating form. Proofs of (a) and (c), using representation theory, appear in [29] and [11]; and in the prime case (b) a proof appears in [8] using methods from Sect. 3. In the following, we embed the collection of all projective s-subspaces of ucker embedding, where n = m+1 P m (Fq ) in P n (Fq ) via the Pl¨ s+1 −1. The image of this embedding is the set Gsn (Fq ) of Fq -rational points of the Grassmann variety Gsn . Recall that Gsn = V(I) where the ideal I ⊂ R is generated by

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certain homogeneous polynomials of degree 2 (the van der Waerden syzygies). The Hilbert function for this variety is known: hI (d) =

0≤j≤s

(m + d − s + j)! j! . (m − s + j)! (d + j)!

Theorem 7. Let G = Gsn (Fq ) = V(I + J) be the set of Fq -rational points of ˆ the Grassmann variety, with n, I, hI as above, and let CG be the Fq -linear code n+1 of length |G| = s+1 q spanned by the hyperplane sections of G. Then dim(CˆG ) = hI (p − 1)r + 1 with hI (d) as above. Note that the Grassmann variety G13 (Fq ) is in fact the Klein quadric, i.e. the hyperbolic quadric in P 5 (Fq ); in this case the dimension of the code CˆG is r

1 p(p + 1)2 (p + 2) + 1, 12 as given by either Theorem 4 or 7.

Acknowledgments The author is pleased to acknowledge the support and encouragement of Professor Bruno Buchberger and the coordinators of the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006) organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz, Austria, for partial support of this project.

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26. G. E. Moorhouse, Ovoids and translation planes from lattices, in N. L. Johnson (ed.) Mostly Finite Geometries, pp. 123–134, Dekker, New York, 1997. 27. G. E. Moorhouse, Some p-ranks related to ﬁnite geometric structures, in N. L. Johnson (ed.) Mostly Finite Geometries, pp. 353–364, Dekker, New York, 1997. 28. G. Panella, Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo ﬁnito, Bull. Un. Mat. Ital., 10 (1955), 507–513. 29. N. Sastry and P. Sin, The codes of generalized quadrangles of even order, in Proceedings of the AMS Summer Research Institute on Group Actions and Cohomology, Seattle, 1996. 30. J. P. Serre, A Course in Arithmetic, Springer, New York, 1973. 31. K. J. C. Smith, On the p-rank of the incidence matrix of points and hyperplanes in a ﬁnite projective geometry, J. Combin. Theory, 1 (1969), 122–129. 32. J. A. Thas, Projective geometry over a ﬁnite ﬁeld, in F. Buekenhout (ed.) Handbook of Incidence Geometry, pp. 295–347, Elsevier, Amsterdam, 1995. 33. J. A. Thas, Ovoids and spreads of ﬁnite classical polar spaces, Geom. Dedicata, 10 (1981), 135–144.

Computer Aided Investigation of Total Graph Coherent Conﬁgurations for Two Inﬁnite Families of Classical Strongly Regular Graphs Matan Ziv-Av Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel. [email protected] Summary. In this chapter we introduce the notion of total graph coherent conﬁguration, and use computer tools to investigate it for two classes of strongly regular graphs – the triangular graphs T (n) and the lattice square graphs L2 (n). For T (n), we show that its total graph coherent conﬁguration has exceptional mergings only in the cases n = 5 and n = 7.

Key words: Triangular graph, Lattice square graph, Total graph coherent closure, Coherent subalgebra

1 Introduction The notion of total graph coherent conﬁguration was introduced and used in [14], where an imprimitive rank 5 association scheme on 40 points was constructed as a merging of relations in the total graph conﬁguration of the triangular graph T (5). In this paper we investigate systematically the total graph coherent conﬁgurations of two inﬁnite series of classical strongly regular graphs. Section 2 contains all preliminaries which make this presentation mostly self-contained. In Sect. 3 we consider the triangular graphs T (n) (for n ≥ 6) and show that in the corresponding total graph coherent conﬁguration T (n) there are two merging association schemes with two and three classes. Besides this there are sporadic mergings for the cases n = 5, 7. Using a computer we prove that these mergings expire all possible merging association schemes in T (n). We also show that T (n) coincides for all n > 4 with the Schurian coherent conﬁguration deﬁned by the automorphism group of the total graph of T (n) (this group is actually Sn , in action on the edges and paths of length 2 of Kn ). In Sect. 4, similar results are presented for the total graph coherent conﬁguration deﬁned by the lattice square graphs L2 (n).

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Our results provide an example of successful amalgamation of essential computer algebra experimentation with subsequent theoretical analysis and generalization. An important feature of presented approach is that the structure constants for both series of coherent conﬁgurations appear as polynomials in variable n. In Sect. 5 we present a detailed outline of the algorithm that we used for the search of mergings, and its implementation in GAP. Finally, in Sect. 6 we discuss further possibilities for investigation of the introduced class of coherent conﬁgurations. Those include relations to the famous graph isomorphism problem and potential applications of Gr¨ obner bases.

2 Preliminaries 2.1 Coherent Conﬁgurations and Association Schemes 2.1.1 Axioms Let X = [1, n], and let R = {R1 , . . . , Rr } be a collection of binary relations on X (subsets of X 2 ) such that: CC1 CC2 CC3 CC4 CC5

Ri ∩ Rj = ∅ for 1 ≤ i = j ≤ r; r 2 i=1 Ri = X ; ∀i ∈ [1, r] ∃i ∈ [1, r]Rit = Ri , where Rit = {(y, x)|(x, y) ∈ Ri }; ∃I ⊆ [1, r] i∈I Ri = Δ, where Δ = {(x, x)|x ∈ X}; ∀i, j, k ∈ [1, r] ∀(x, y) ∈ Rk |{z ∈ X|(x, z) ∈ Ri ∧ (z, y) ∈ Rj }| = pkij ,

then M = (X, R) is called a coherent conﬁguration. The relations in R are the basis relations of M. The parameters pkij are the structure constants of the conﬁguration. The graphs Γi = (X, Ri ) are called the basis graphs of the coherent conﬁguration. See [11] for the original deﬁnitions. Let (G, Ω) be a permutation group. G acts naturally on Ω 2 by (x, y)g = g g (x , y ). Following H. Wielandt in [18], the orbits of this action, (G, Ω 2 ) are called the 2-orbits of (G, Ω), denoted by 2-orb(G, Ω). For every permutation group (G, Ω), (Ω, 2-orb(G, Ω)) is a coherent conﬁguration. Conversely, if the set of relations of a coherent conﬁguration, M, coincides with 2-orb(G, Ω) for a suitable group G, then M is called a Schurian coherent conﬁguration. A coherent conﬁguration which has Δ = {(x, x)|x ∈ X} as one of its basis relations is called an association scheme. In this case, all basis relations except for Δ are called classes. A fusion conﬁguration (or a merging) of a coherent conﬁguration M = (X, R) is a coherent conﬁguration M = (X, S) on the same set such that each basis relation Si of M is a union of basis relations of M. Coherent conﬁgurations can be alternatively described in matrix language. The adjacency matrix A(R) of a relation R on X is a (0, 1)-matrix A(R) =

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(aij ) of dimension |X|×|X| such that aij = 1 iﬀ (i, j) ∈ R. If R = {R1 , . . . , Rr } are the basis relations of a coherent conﬁguration then their adjacency matrices {Ai = A(Ri )}ri=1 form a basis of a matrix algebra which is closed under Schur-Hadamard (element wise) product. This leads to equivalent formulation of the axioms of coherent conﬁguration: Let W ⊆ Cn×n be a matrix algebra of square matrices of order n over the complex ﬁeld, such that CA1 Was a linear space over C has some basis, A1 , A2 , . . . , Ar , consisting of (0, r1)-matrices; CA2 i=1 Ai = Jn , where Jn is the square matrix of order n all entries of which are equal to 1; CA3 ∀i ∈ [1, r]∃i ∈ [1, r]Ati = Ai ; CA4 I ∈ W (I denotes the identity matrix), then W is called a coherent algebra of rank r and order n with the standard basis C = {A1 , A2 , . . . , Ar }. We write W = A1 , . . . , Ar . The notion corresponding to a fusion scheme in this notation is a coherent subalgebra, that is a subalgebra which is also a coherent algebra. 2.1.2 Weisfeiler-Leman Closure Using matrix notation, it is easy to see that the intersection of coherent algebras is a coherent algebra, and that each square matrix is contained in some coherent algebra (since Mn (C) is coherent). Therefore, we can deﬁne the coherent closure of a matrix A, denoted A

as the smallest coherent algebra containing this matrix (or in other words, the intersection of all coherent algebras containing it). An eﬃcient algorithm for computing A

was suggested by Weisfeiler and Leman [17] and is frequently called the WL-stabilization of the matrix A. 2.1.3 Wreath Product If M1 = (X1 , {R0 , R1 , . . . , Rr−1 }) and M2 = (X2 , {S0 , S1 , . . . , Sl−1 }) are association schemes (R0 and S0 are the reﬂexive relations), then the wreath product of M1 with M2 is deﬁned as M1 M2 = (Y = X1 × X2 , {T0 , T1 , . . . , Tr+l−2 }) where T0 is the identity relation on Y , Ti = {((a, b), (c, d))|(a, c) ∈ Ri } for all 1 ≤ i ≤ r − 1, and Tr−1+i = {((a, b), (a, c))|(b, c) ∈ Si } for all 1 ≤ i ≤ l − 1. The wreath product of association schemes of ranks r and l is an association scheme of rank r + l − 1. 2.2 Total Conﬁguration Let Γ = (V, E) be a graph. The total graph T (Γ ) is the graph with the vertex set V ∪ E, two such vertices in T (Γ ) are adjacent if and only if they are

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adjacent or incident in Γ (here edges of Γ are incident if they have a joint vertex). The coherent closure of T (Γ ) will be called the total coherent conﬁguration of Γ . The Schurian total coherent conﬁguration of a graph Γ is (X, 2-Orb(Aut(T (Γ )))) where X is the set of vertices of T (Γ ). The total conﬁguration is a fusion of the Schurian total conﬁguration. Indeed, since an automorphism of T (Γ ) maps edges of T (Γ ) to edges, any 2-orbit of Aut(T (Γ )) either contains only edges, or does not contain edges at all. 2.3 Computational Tools 2.3.1 COCO COCO is a set of programs for dealing with coherent conﬁgurations. The current version was developed during the years 1990–1992 in Moscow, USSR, mainly by Faradˇzev and Klin [8, 7]. COCO can be used to construct a coherent conﬁguration (Ω, 2-orb(G, Ω)) from a prescribed permutation group (G, Ω), as well as to calculate the structure constants of the constructed coherent conﬁguration, and ﬁnd all association schemes which are mergings of the coherent conﬁguration, together with their automorphism groups. COCO was originally written for DOS, and the version currently in use is the UNIX port by A.E. Brouwer, available from Brouwer’s home page [4]. 2.3.2 WL-stabilization Two implementations of the Weisfeiler-Leman stabilization [17] are available, under the name stabil [1] and stabcol [2]. The two implementations diﬀer slightly in memory usage and run time, but both are adequate for the coherent conﬁgurations used in this article. 2.3.3 GAP GAP [9, 16], an acronym for “Groups, Algorithms and Programming”, is a system for computation in discrete abstract algebra. The system is extensible in the sense that it supports easy addition of extensions (packages, in GAP nomenclature), that are written in the GAP programming language which can extend the abilities of the GAP system. Within GAP framework, COCO-II (a reimplementation of COCO functionality as a GAP package, currently in development by S. Reichard et al.) will be used. COCO-II improves on the original COCO by adding functionality such as WL-stabilization, as well as using algorithms developed since the release of COCO. The author modiﬁed some COCO-II functions to handle polynomial structure constants, instead of the usual numeric constants, and those functions are used to handle the general case in this paper.

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3 Total Conﬁguration of Triangular Graph 3.1 Deﬁnition and Basic Properties Let T(n) denote total graph of the triangular graph T (n) (recall that T (n) is the line graph L(Kn ) of the complete graph Kn ), and let T (n) denote the coherent closure of T(n). In more detail, the total graph T(n) is the total graph of the triangular graph T (n). The vertices of T(n) are the edges and the paths of length 2 of Kn . The edges of T(n) are partitioned to three types: {e, f }, where e and f are edges of Kn which share a common point. {e, P }, where e is an edge of Kn , P is a path of length 2 in Kn , and P includes e. {P, Q}, where P and Q are paths of length 2 in Kn which share a common 2 vertices. edge. T(n) has n(n−1) 2 To investigate T (n) we will ﬁrst consider the Schurian total coherent conﬁguration S(n), and later show that T (n) = S(n). Let Ω be the set of vertices of T(n). In other words, Ω = {{a, b}|a = b ∈ [1, n]}∪{{{a, b}, {a, c}}|a, b, c ∈ [1, n], a = b, a = c, b = c}. Let G be the automorphism group of T(n); it is well known that G is the permutation group (Sn , Ω) (with the natural action of Sn on Ω) for n > 4. Then, S(n) = (Ω, 2 − orb(G)) is the Schurian total coherent conﬁguration of the triangular graph. For n ≥ 6, S(n) has 2 ﬁbres and 25 relations as follows (for the sake of brevity, we will list a standard compact description for each relation), see also Table 1. Two reﬂexive relations: R0 = ({a, b}, {a, b}), Table 1. Sizes of basis relations of S(n) Relation 0 2 4 6 8 15 17 19 21 23

Size n 2

n(n 2) n − 1)(n− n−3 (n − 2) 2 n2 n2 2(n − 2)(n − 3) − 2) 2 (n n−3 nn−1 2 2 n−1 n 2 2 n−3 2 2(n − 3) nn−1 2 2(n − 3) nn−1 2 2(n − 3) n n−1 2

Relation 1 3 5 7 14 16 18 20 22 24

Size n 3 3

n n−2 2 n2 n−2 n2 2 2 2 2(n − 2) n−4 nn−1 (n − 3) 2 2 nn−1 (n − 3)(n − 4) 2 nn−1 2(n − 3)(n − 4) 2 nn−1 2(n − 3) 2 nn−1 (n − 3) 2 n−1 n

2

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R1 = ({{a, b}, {a, c}}, {{a, b}, {a, c}}). Two relations within ﬁrst ﬁbre: R2 = ({a, b}, {a, c}) (arcs of the triangular graph), R3 = ({a, b}, {c, d}) (arcs of its complement), ﬁve relations between ﬁrst and second ﬁbre: R4 = ({a, b}, {{c, d}, {c, e}}), R5 = ({a, b}, {{a, c}, {a, d}}), R6 = ({a, b}, {{c, a}, {c, d}}), R7 = ({a, b}, {{a, b}, {a, c}}), R8 = ({a, b}, {{c, a}, {c, b}}), and the ﬁve inverses R9 , . . . , R13 respectively, eleven relations within second ﬁbre: R14 = ({{a, b}, {a, c}}, {{d, e}, {d, f }}), R15 = ({{a, b}, {a, c}}, {{a, d}, {a, e}}), R16 = ({{a, b}, {a, c}}, {{d, a}, {d, e}}), R17 = ({{a, b}, {a, c}}, {{b, d}, {b, e}}), R18 = ({{a, b}, {a, c}}, {{d, b}, {d, e}}), R19 = ({{a, b}, {a, c}}, {{a, b}, {a, d}}), R20 = ({{a, b}, {a, c}}, {{b, a}, {b, d}}), R21 = ({{a, b}, {a, c}}, {{d, a}, {d, b}}), R22 = ({{a, b}, {a, c}}, {{d, b}, {d, c}}), R23 = ({{a, b}, {a, c}}, {{b, c}, {b, d}}), R24 = ({{a, b}, {a, c}}, {{b, a}, {b, c}}). (Note, that of the last 11 relations, 14, 15, 18, 19, 20, 22 and 24 are symmetric, (16, 17) (21, 23) are the anti-symmetric pairs.) The set of edges of the total graph of the triangular graph is the union R2 ∪ R7 ∪ R12 ∪ R19 ∪ R20 ∪ R24 . 3.2 Structure Constants of S(n) Proposition 1. The structure constants of S(n) are functions of n. For n ≥ 9, each such function is a polynomial function in n of degree at most 3. Proof. To calculate a structure constant pki,j , we take the general representative pair, (X, Y ), of Rk , and try to ﬁnd a general vertex Z such that (X, Z) ∈ Ri and (Z, Y ) ∈ Rj . The selection of (X, Y ) partitions the points [1, n] into at most 6 parts of constant size (not dependent on n, but dependent on X, Y ), and one part of size n − k where k is again dependent on X, Y but not on n. So, the number of ways of selecting Z is a product of two or three of the sizes of parts, or sizes of parts minus one, or sizes of parts minus 2 (or maybe half this product, in case a set of two points needs to be selected), and therefore is a polynomial of degree at most 3 in n.

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Examples: If we want to calculate p15 10,5 : ({{a, b}, {a, c}}, {{a, d}, {a, e}}) partitions the set [1, n] to parts {a}, {b, c}, {d, e}, and the rest, of size n − 5. A vertex {x, y} such that ({{a, b}, {a, c}}, {x, y}) is in R10 and ({x, y}, {{a, d}, {a, e}}) is in R5 must satisfy x ∈ {a}, y ∈ {a, b, c}, x ∈ {a}, y ∈ {a, d, e}. We have one way of selecting x and n − 5 ways of selecting y, so p15 10,5 = n − 5. If we want to calculate p14 14,14 : ({{a, b}, {a, c}}, {{d, e}, {d, f }}) partitions [1, n] into {a}, {b, c}, {d}, {e, f }, and we need to ﬁnd {{x, y}, {x, z}} such that x, y, z ∈ {a, b, c, d, e, f }, ways to select y and z, so so we have n − 6 ways to select x and (n−7)(n−8) 2 (n−6)(n−7)(n−8) 14 p14,14 = . 2 This last example shows why we need to assume n ≥ 9 for the general argument. This is the case that requires the maximal number of points from the original graph Kn . Now we can use a computer to calculate the actual polynomials: Using COCO, we ﬁnd numerical values for all structure constants in the cases n = 9, 10, 11, 12. Then for each triplet i, j, k we use Lagrange interpolation in GAP to ﬁnd the polynomial pki,j (n). 3.3 S(n) and T (n) Proposition 2. S(n) = T (n). Proof. First we recall that the total graph T(n) is the union of the relations R2 ∪ R7 ∪ R12 ∪ R19 ∪ R20 ∪ R24 , so T (n) is a fusion conﬁguration of S(n). It is enough to show that no proper fusion conﬁguration of S(n) contains T(n). For 4 < n < 9 we check by a computer implementation of the WeisfeilerLeman closure algorithm that the closure of the total graph is indeed the conﬁguration S(n) (without the empty relation R14 in the case n = 5). By using COCO, we ﬁnd that for n ≥ 6 there are two fusion association schemes of ranks 3 and 4, except for n = 7 where there is another scheme of rank 3. For n ≥ 9, we conﬁrm by computer search that those two mergings always appear, no other mergings appear for all n, and that there are no sporadic mergings other then those described in 3.4. This proves that S(n) = T (n), since T (n) is a merging of S(n), but the mergings we found do not admit T(n) as a union of relations. 3.4 Mergings of S(n) We are looking for mergings of S(n) (for n ≥ 6) resulting in association schemes. The result of the search as described in Sect. 5 is that in the general case (that is, for n ≥ 6) there are only two mergings: 2 , n − 2, n − 3, 0). A strongly regular graph, Γ , with parameters ( n(n−1) 2 This SRG is the union of the relations R8 ∪ R13 ∪ R22 . This graph can be

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deﬁned on the vertices of T(n), denoted by {a, b} and ({a, b}, c) (the latter standing for {{a, c}, {b, c}}), as follows: two vertices are adjacent if they share copies of the same two-set. Since μ = 0, this graph is isomorphic to n(n−1) 2 Kn−1 . A rank 4 association scheme, whose classes are the unions of the relations: R0 ∪ R1 ; R8 ∪ R13 ∪ R22 ; R2 ∪ R6 ∪ R7 ∪ R11 ∪ R12 ∪ R18 ∪ R19 ∪ R21 ∪ R23 ∪ R24 ; R3 ∪ R4 ∪ R5 ∪ R9 ∪ R10 ∪ R14 ∪ R15 ∪ R16 ∪ R17 ∪ R20 . When using the above notation ({a, b} and ({a, b}, c)) for the set of vertices of T(n), the relations of this merging association scheme can be deﬁned by the number of points their two-sets share. With this observation we recognize that the rank 4 scheme is the wreath product MKn−1 , where M is the rank 3 association scheme with basis graphs Δ, T (n), and T (n). The only exception is for n = 7 which has another SRG as a merging of the relations: R3 ∪ R4 ∪ R8 ∪ R9 ∪ R13 ∪ R15 ∪ R18 ∪ R20 . This SRG has parameters (126, 45, 12, 18), and will be discussed in subsequent publication [13]. For n = 5, relation R14 is actually empty, since it requires 6 diﬀerent points of Kn . So S(5) is a rank 24 coherent conﬁguration. This conﬁguration has 9 merging association schemes listed in Table 2. (All mergings also merge the reﬂexive relations R0 and R1 .) Some of the mergings are discussed in [14, 13].

4 Total Conﬁguration of L2 (n) 4.1 Deﬁnition and Basic Properties The lattice square graph, L2 (n), is a graph with n2 vertices, usually the vertex set is denoted by [1, n]2 , with (a, b) adjacent to (c, d) if a = c or b = d. It is useful to regard the vertices as the points of the n × n-grid. Two vertices are adjacent if and only if they are in the same row or the same column. For our purposes it is also useful to see this graph as the line graph of the complete bipartite graph Kn,n . L2 (n) is a regular graph of valency 2(n − 1), so it has n2 (n − 1) edges. The total graph T (L2 (n)) has n2 + n2 (n − 1) = n3 vertices. Let us denote the vertices of L2 (n) by (a, x), where a, x ∈ [1, n]. The automorphism group G = Aut(L2 (n)) of order 2(n!)2 is generated by Sn acting on ﬁrst coordinate, Sn acting on second coordinate, and involution mapping (a, x) to (x, a), denoted by t. In other words, G is the exponentiation Sn ↑ S2 , as in [8]. In this notation, edges of L2 (n) are of the form {(a, x), (a, y)} (a pair of vertices in the same row) or {(x, a), (y, a)} (a pair of vertices in the same column), here x = y.

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Table 2. Mergings of S(5) Rank 5

Mergings (3, 4, 5, 9, 10, 15, 16, 17, 20) (8, 13, 22) (2, 7, 12, 18, 19, 24) (6, 11, 21, 23)

|aut|

SRG parameters

1920

5

(3, 4, 9, 15, 20) (8, 13, 22) (5, 10, 16, 17) (2, 6, 7, 11, 12, 18, 19, 21, 23, 24)

7680

5

(3, 7, 8, 12, 13, 15, 16, 17, 18) (4, 9, 24) (2, 5, 10, 19, 20, 22) (6, 11, 21, 23)

1920

5

(3, 8, 13, 15, 18) (4, 9, 24) (2, 5, 6, 10, 11, 19, 20, 21, 22, 23) (7, 12, 16, 17)

7680

4

(3, 4, 5, 9, 10, 15, 16, 17, 20) (8, 13, 22) (2, 6, 7, 11, 12, 18, 19, 21, 23, 24)

233 311 5

4

(3, 7, 8, 12, 13, 15, 16, 17, 18) (4, 9, 24) (2, 5, 6, 10, 11, 19, 20, 21, 22, 23)

233 311 5

3

(2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24) (8, 13, 22)

238 314 52 7

(40, 3, 2, 0)

3

(2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22, 24) (6, 11, 21, 23)

51,840

(40, 12, 2, 4)

3

(2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23) (4, 9, 24)

238 314 52 7

(40, 3, 2, 0)

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We shall denote the total coherent conﬁguration of L2 (n) by T(n) and the Schurian total conﬁguration of L2 (n) by S(n). In the following listing of representatives of relations of S(n), a, b, c, d stand for distinct elements of [1, n], and x, y, z, w stand for distinct elements of [1, n]. The sets {a, b, c, d} and {x, y, z, w} are not necessarily disjoint. In relations R4 , . . . , R11 the representative of edges appear all as a pair of vertices in the same row. Since the involution t is in the automorphism group, and maps a pair of vertices in the same row to a pair of vertices in the same column, those edges are also represented. For example, ((a, x), {(a, x), (b, x)}) is in R4 , since it is the result of action of t on ((x, a), {(x, a), (x, b)}) which is clearly in R4 . In the same manner, when looking at relations R12 , . . . , R20 (pairs of edges of L2 (n)), it does not matter if the ﬁrst edge is a pair of vertices in the same row or a pair of vertices in the same column, but it does matter whether both edges are of the same kind (row or column) or of diﬀerent kinds. Relations R12 , . . . , R16 are of the former type, while relations R17 , . . . , R20 are of the latter type. S(n) has the following 21 relations (for n ≥ 3), see also Table 3. Reﬂexive relations: R0 = ((a, x), (a, x)), R1 = ({(a, x), (a, y)}, {(a, x), (a, y)}). Relations within ﬁrst ﬁbre: R2 = ((a, x), (a, y)), R3 = ((a, x), (b, y)). Relations between ﬁrst and second ﬁbre: R4 = ((a, x), {(a, x), (a, y)}), R5 = ((a, x), {(b, x), (b, y)}), R6 = ((a, x), {(a, y), (a, z)}), R7 = ((a, x), {(b, y), (b, z)}). Their inverses: R8 = ({(a, x), (a, y)}, (a, x)), R9 = ({(b, x), (b, y)}, (a, x)), R10 = ({(a, y), (a, z)}, (a, x)), R11 = ({(b, y), (b, z)}, (a, x)). And relations within second ﬁbre: R12 = ({(a, x), (a, y)}, {(a, x), (a, z)}), R13 = ({(a, x), (a, y)}, {(a, z), (a, w)}), R14 = ({(a, x), (a, y)}, {(b, x), (b, y)}), R15 = ({(a, x), (a, y)}, {(b, x), (b, z)}), R16 = ({(a, x), (a, y)}, {(b, z), (b, w)}), R17 = ({(a, x), (a, y)}, {(a, x), (b, x)}), R18 = ({(a, x), (a, y)}, {(a, z), (b, z)}), R19 = ({(a, x), (a, y)}, {(b, x), (c, x)}), R20 = ({(a, x), (a, y)}, {(b, z), (c, z)}). R18 and R19 form an anti-symmetric pair. All other relations within second ﬁbre are symmetric. The total graph T (L2 (n)) is the union of the relations: R2 ∪ R4 ∪ R8 ∪ R12 ∪ R17 . 4.2 Structure Constants of S(n) As in the case of the triangular graph in the previous section, when we calculate pkij for a given triplet (i, j, k), we actually take an element (M, N ) of relation Rk , and count the amount of elements P such that (M, P ) ∈ Ri and (P, N ) ∈ Rj . Here P is either a vertex or an edge of L2 (n), so we need to

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Table 3. Sizes of basis relations of S(n) Relation 0 2 4 6 12 14 16 18 20

Size n2 2n2 (n − 1) 2n2 (n − 1) n2 (n − 1)(n − 2) 2n2 (n − 1)(n − 2) n2 (n − 1)2 1 2 n (n − 1)2 (n − 2)(n − 3) 2 2 n (n − 1)2 (n − 2) 1 2 n (n − 1)2 (n − 2)2 2

Relation 1 3 5 7 13 15 17 19

Size n2 (n − 1) n2 (n − 1)2 2n2 (n − 1)2 n2 (n − 1)2 (n − 2) 1 2 n (n − 1)(n − 2)(n − 3) 2 2n2 (n − 1)2 (n − 2) 2n2 (n − 1)2 n2 (n − 1)2 (n − 2)

select two or three elements of [1, n]. For each element of [1, n] that we need to select, it either needs to be one already used in M or N , in which case the number of options is a constant independent of n, or not used, in which case the number of options is n − r, where r is dependent on i, j, k, but not on n. After all selections, we might need to multiply by 2 (if the representative element of Ri is not invariant under the involution t), and similarly for Rj an Rk . We also need to divide by 2, if P is and edge of L2 (n), since we selected two elements the order of which is irrelevant. Finally, we conclude that pkij is a polynomial function in n of degree at most 3. For ﬁnding the minimal n for which this argument will work, we note that the worst case is in the calculation of p16 16,16 , where we have a pair ({(a, x), (a, y)}, {(b, z), (b, w)}), and need to ﬁnd an edge of L2 (n) {(c, u), (c, v)}, such that u is diﬀerent from x, y, w, z, and so is v. In conclusion: Proposition 3. The structure constants of S(n) are functions of n. For n ≥ 6, each such function is a polynomial function in n of degree at most 3. 4.3 Mergings of S(n) S(n) admits no association schemes as mergings (for n ≥ 3). 4.4 S(n) and T(n) Proposition 4. S(n) = T(n) Proof. First we recall that the total graph T (L2 (n)) is the union of the relations R2 ∪ R4 ∪ R8 ∪ R12 ∪ R17 . So, T(n) is a fusion conﬁguration of S(n). It is enough to show that no proper fusion conﬁguration of S(n) contains T (L2 (n)). This is done by computer search.

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5 Details of Computer Search The computer search mentioned in Sects. 3.3 and 4.4 is based on the notion of good sets, which goes back to [8]: If W = A0 , . . . , Ar is a coherent algebra, then we deﬁne a good set to be a subset B ⊆ [0, r] such that: GS1 M = i∈B Ai is an adjacency matrix of a symmetric or an anti-symmetric relation; r GS2 if M 2 = i=0 bi Ai then for every i, j ∈ B, bi = bj ; GS3 I ◦ M = M or I ◦ M = 0 (◦ is Schur-Hadamard product). With this deﬁnition of a good set, we see that for a partition P = {P1 , . . . , Pk } of [0, r] to induce a coherent subalgebra, each Pi must be a good set. This reduces the computational search for subalgebras from a search through all partitions of [0, r], to a search through partitions consisting of good sets only. This method, originally developed in [8] for use with a numerical tensor of structure constants, also works for a polynomial tensor. A set that is good by its polynomial parameters, that is good for all n ≥ 9, is called polynomially good set. I The graphs T(n) are constructed for n = 9, 10, 11, 12, and for each such n, the WL-closure, T (n) is calculated. We then check that it coincides with S(n), which is calculated by COCO. II The structure constants of the four coherent conﬁgurations are used to generate the polynomial tensor of structure constants (using Lagrange interpolation). III Instead of searching for all mergings, we limit our search to speciﬁc kinds of mergings: i Mergings resulting in association schemes: Since the basis graphs of an association scheme are regular, we add another requirement for a good set: the graph with edge set i∈B Ri must be regular. The number of good sets with this additional condition is 5, and a quick search shows that only two mergings (those described in 3.4) appear. ii Mergings that admit T (n) as a merging. This means that Q = {2, 7, 12, 19, 20, 24} is a union of sets from the partition, or in other words an additional condition for a good set B is that either B ⊆ Q or B∩Q = ∅. Since none of the previous two mergings fulﬁll this condition, we know that such mergings do not result in association schemes. Since we only have two ﬁbres, the mergings we are looking for also have two ﬁbres. This allows us to partition the relations into cells according to the ﬁbres: {{0}, {1}, {2, 3}, {4, 5, 6, 7, 8}, {9, 10, 11, 12, 13}, {14, . . . , 24}} and require a good set to comply with partition, that is, to be a subset of one of the sets in the partition.

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Those two conditions leave 48 good sets and a simple search shows that none of the partitions result in a coherent conﬁguration. IV The previous step is enough to show that there is no merging (except for the two described in 3.4) that appear for every n ≥ 9. To conﬁrm that no other mergings appear for particular n > 7 we use the following principle: When we check if the set B = {a1 , . . . , al } is good, we actually calculate the sums Qk = i,j∈B pki,j for each k ∈ B. Clearly, if all these sums are equal, then the set is good. If it is not good, then we have (at least) two elements i, j ∈ B such that the polynomials Qi and Qj diﬀer. If a natural number n0 is a root of the polynomial Qi − Qj , it means that while the set B is not a good set for all n, it might be a good set for n0 . In this case we add this n0 to the list of n for which an exceptional merging might appear. In the case of S(n), the list of possible exceptions included all the integers in the range [1, 22], and a computerized brute force search shows that for no n in the range 7 < n ≤ 22 does an exceptional merging appear. A similar search is performed for the total conﬁgurations of the lattice square graphs. In this case there were no mergings resulting in association schemes which appear for all n, and the list of possible exceptions is {3, 4, 5, 6, 7, 8, 9, 10, 11, 17}. A computerized search shows that there is no exceptional merging resulting in an association scheme for any of those values.

6 Conclusions It is easy to check that the total graph of the complete graph Kn with n vertices is isomorphic to the triangular graph T (n + 1). This observation by [3], see also [10] was one of the earliest stimuli of interest in this concept. We again refer to [14] for a detailed discussion of exceptional Schurian association scheme with the automorphism group of order 1920 which appears as a merging of classes in T (5). The triangular graph T (5) is a complement of the Petersen graph. Petersen graph may be considered as the smallest Moore graph (see, e.g. [5]). This is why we are also investigating the total graph conﬁgurations of the complements of the unique Moore graph of valency 7 and of a potential Moore graph of valency 57, see [13]. It seems as a very attractive task to search for other examples of strongly regular graphs, which have total coherent conﬁguration admitting exceptional mergings. Our results for the graphs L2 (n) provide a small evidence for believing that such examples are quite rare. We hope that the use of Gr¨ obner bases (cf. [15]) may help in computerized investigation of other inﬁnite series of classical strongly regular graphs. The problem of the description of the 2-orbits of the automorphism group of an arbitrary graph Γ is closely related to the graph isomorphism problem,

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see e.g. [12]. In case when for a graph Γ its total graph coherent conﬁguration and the Schurian total graph coherent conﬁguration coincide, using WL-closure, we get as a by-product a polynomial-time procedure for the description of 2-orb(Aut(Γ )). At this moment we are not aware of an example where those two coherent conﬁgurations are distinct for strongly regular graphs. Certain counterexamples for arbitrary graphs will be discussed in [13], in particular, in view of the results from [6] and related publications.

Acknowledgments This project is a part of the graduate thesis which the author is writing under the supervision of M. Klin. The author is pleased to acknowledge Prof. Bruno Buchberger and the coordinators of the Special Semester on Gr¨ obner Bases (February 1 – July 31, 2006), organized by RICAM, Austrian Academy of Sciences, and RISC, Johannes Kepler University, Linz Austria. The conceptualization of this text goes back to the time of workshop D1 at Linz, May 2006, which was in the scope of the Special Semester.

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